Euclide's Elements; the whole fifteen books compendiously demonstrated. With Archimedes Theorems of the sphere and cylinder, investigated by the method of indivisibles

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•

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•

E LI C L I D E v

ELEMENTS;

The whole Tifteen BOOKS

Compendioufly Demonftrated.

WITH

Archimedes Theorems

Of the Sphere and Cylinder, invefti-
gated by the Method of Indivifibles.

By ISAAC BARROW, D. D. Late Majler

of Trinity College in Cambridge.

To which is added in this Edition,

EVCLIDE's D ATA
with Marinus's Preface.

And a Brief TREATISE of

REGVLAK SOLIDS.

€ ■

10 ifDOH: Printed and Sold by W. Redmayne
in Jewcn-ftret$f X* Mount on Tower- billy and J. and
£. Sprint in LittMritain. 1714.

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To the READER.

f>« mdefimss, Courteous Reader f o know what
Ibave perform J in this Edition of the Elements
of Evclide, I fii*U here explain it to youtn jhort,
48. according to the nature of the Work. Ibave en-
deavor* 1 to attain two end, chief, J the fir f, to be very
ferfpitaous, and at the fame time fo very bnef, that,
the Book may not [mil to fuch
trouble {owe to carry about one, in which 1 1 fink I bave
fuccceded, unlefs in my abfence theP r inter scare fsould
frufiratemy Dcfign. Some of a brighter Genius and
endued with greater Skill, may have demonfratedmofi
of thefe Proptfitions with more nicety, but perhaps none
With more fuccinanefs than I have J
alter d nothing in the number and order of the Authors
Propofitions ; %>r prefum'd either Jo take the liberty of

fimc of the eafafct into the rank of ' Jxum,
veralhave <hne , and among others that mofieXpe*
Geometrician A. Tacquetus C (whom I the™
willingly namt) becaufe I think U u hut
knowledge that I have imitated him m feme Points)
after Afe mo/l aerate Ednion I had no foughn of
attempting any thing of this nature, tdl I ™ lder f
that \his*mof leaned M*n thought fit to pshlifson.y
Sight ,f Euclide'i Bocks, which he took the pains t
4lainandembell,(li} having VtA^^W
Jd undervalued the other feven, as lefs nppe, -tasmng
u the Elements of Geometry. But my Evince w«

a 2 91 *

a

r.

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To the READER:

eriginally quite different 9 not that of writing the Ele-
ments o f Geometry 9 after what method foever I pleas d,
but of demon ftrating, in as few words as poffible 1
coud, the whole Works of Euclide. As to four of
the Books y viz. thefeventby eighth y ninth, and tenth ,
altho they dont fo nearly appertain to the Elements of
plain and folid Geometry, as the fix precedent and the
two fubfeyuent, yet none of the more skilful Geome-
tricians can be fo ignorant as not to know that they are
very ufeful for Geometrical matters y not only by rca+
fon of the mighty near affinity that is between Arith-
metick and Geometry 3 but alfo for the knowledge of
both mea fur able and unmeafurable Magnitudes9 fo ex*
eeeding neccffary for the Dottrine of both plain and fo-
lid figures. Now the noble Contemplation of the five
Regular Bodies that is contain d in the three lafi Books ,
cannot without great Injufiice be pretermittedy Jince
that for the fake thereof our wyuurtK, being a Pbi-
lofopbertf the Platonic Sefty is f aid to have eomposd
this univerfal Syjlem of Elements ; as Proclus lib. 2.
Witneffetb in thefe Wordsy *08ir H *) tit* ev^dcni

icovik&v %y\h&7wv obs&aiv. Be fides y I eafily perfwaded
my felf to think, that it would not be unacceptable
to any Lover of thefe Sciences to have in his PoffeJJion
the whole Euclidean Worky as it is commonly cited
and celebrated by all Men. Wherefore lrefolvdto
omit no Book or Proportion of tbofe that are found in •
P. Herigonius'i Edition, wbofe Steps I was obliged
elofely to follow, by reafon I took a Refolution to make
ufe of moft of the Schemes of the faid Booky very
well fore feeing that time would not allow me to form
• , • new

*

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To the READERS

new ones, tho fometimes I chofe rather to do it. ' Far
the fame reafon I was willing to ufefdr the mofi fart
Euclide'* own Demon ftrations, having only exprefsd
them in a more fuccintt Form, unlefs Perhaps in tit
fecondy thirteenth, and very few in the fcventb, eighth ±
and ninth Book, in which it feemd not worth my
while to deviate in any particular from him. There" •
fore I am not without good hopes that as to this part I
have in fome meafure fatisfied both my own Intentions,
and the Dejire of the Studious. As for fome certain
Problems and Theorems that are added in the Scbolions
(or fltort Expo/itions) either appertaining (by reafon of
their frequent Vfe) to the nature ofthefe Elements, or
conducing to the ready Demonjlration of tbofe things
that follow, or which do intimate the reafons of fome
principal Rules of practical Geometry, reducing them
to their original Fountains, thefe I fay, will not, I
hope, make the Book fwell to a Siz>c beyond the dc+
fignd Proportion.

The other Butt, which I levell'd at, is to content
the De fires of tbofe who are delighted more with fym-
bolical than verbal Dcmonfirations. In which kind,
whereas mofi among us are accufiomd to the Symbols
i/Gulielmus Oughtredus, I therefore thought befi
to make ufe, for the mofi part, of bis. None hitherto
(as I know of) has attempted to interpret and publics
Euclide after this manner, except P. Herigonius ;
wbofe Method ( tho indeed mofi excellent in many
things, and very well accommodated for the particular
purpofe of that mofi ingenious Man) yet feems in my
Opinion to labour under a double DefeB. Firfi, in
regard that, altho of two or more Propofitions, produ-

\ ceJ

y Google

To the READER:

ced for the Proof of any due Problem or Theorem, the
former don t always depend of the latter, yet it don't
readily enough appear either from the order of each, or
by any other manner, when they agree together, and
when not k j wherefore for want oftbeConjunttiuns and
AdjcBives, ergOj riirfus, &e. many difficulties and
*cccafions of doubt do often arife in reading, cfpecially
to tbsfe that are Novices. Befides it frequently hap-
pens, that the faid Method cannot avoid fuperflttout
Repetitions, by which the bemonftratiens are often-
times render d tedious, and fometimes alfo more intri*
cate ; which Faults my Method doth eafily remedy by
the arbitrary mixture of both Words and Signs. There-,
fore let what has been faid, torching the Intention
and Method of this little Work, fuffice. As to the.
reft, whoever covets to pleafe himfelfwith what may
be faid, either in Praife of the Mathematicks in gene-
ral, or of Geometry in particular, or touching the Hi-
ftory of thefe Sciences, and consequently of EucHde
himfelf (who digefted thofe Elements) and others
ifyriVKX of that kind, may cenfult other Interpreters.
Neither will I {as if I were afraid leaji thefe my En-
deavors may fall Jhort of being fatisfaBory to all Per-
sons) alledge as an Excufe (tho I may very lawfully
do it) the want of due time which ought to be em-i
ploy d in this Work, nor the Interruption occajiond by
other Affairs, nor yet the want of reauifite help for
thefe Studies nor fever al other things of the like nature.
But what I have here employ d my Labour and Study
in for the Ufe of the ingenuous Reader, I wholly
fubmit to his Cenfure and Judgment, to approve if
vfefuL or rejeft if of her wife.

. J . J. B.

4

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Ad aroiciffimum Vlmm, i., G <le NUCLIDE
contra&o, Eilfttft/^i^.

Jp^#r/»/ Jfiie/ didicit Laconice loqui

A&ftr profundus, aphonfmos induX.
Immenfa dudum margo comment aril
Diagtamma circuit minutum ; irtjt/e Jr/ttk
Vroblema breve natahat in vafio mari.
Sed unda jam detumuit\ & glojfa arSior
Stringk Theoremata : minoris anguli
Lateribus ecce tofw/EucIidw jnvet,
Inclufus ohm velut Hornems in nuce ;
Pluteoque farcina modo qtri mtidfttrt, levis
En fit manipulus. Velle in exigua latet
Ingens MatJjeJls, matris utero fi amies ,
Inglande quercus, vel Itliaca Bums in pilct<

1

I

Nec mole dum iecrefcit> ufu fit minor \
Am au&iar jam evadit, £f cumulates

rr

ContiaSa prodejt emdita pngma>
Sic ubere magis liquor e pejfo ajfiuit \
Sic pleniori vafa imcndat fangumis
Torrente cordis Sy]tole\ ficfujms
Procurrit <zquw ox AbyU angujtiis.
TantiUi operrs ars taittn referenda vnice eft
BAROVIANO nominiy ac folertia. t
Sublimis euge mentis ingertrumyotens 1
Cui invitum nil, arduum ejfe nilfolet$
Sic ufque pergas profpero conarmnef
Radiufque multum Jebeat ac abacus tibi 5 I
Sic ere/cat indies feracior fegesy
Simili colonum genuine ajjithto beans.
Specimen future mejjis he fict labor.
Magn&que fmns. iUuJh ia %ac pysUidicu
Juvcnis dedlp $ui t*uta> quid dabit fenex ?

' Car. RoixKhtm, CJNIJ£.
Coll. Inn* S&j. Sec.

*

The

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The Explication of the Signs- or

Characters.

fEqual.

Greater.
I (.efler. .
More, , or to be added.
Lets, ot to be fubtrafted.

V

Q&q
C&C

Q..Q.

or Escefs ; Alfo, that all
the quantities which follow, are to be
3 I fubtrafted, the Signs not being changed.

... '.v

.§> Multiplication, 01 the Drawing one fide of
w . a Redande into another.

ae j

as

ThcSidc orl^ootof aSquare,
A Square. I
A Cube.

The {Suae is denoted by the Conjuration

^

Offcr Abbreviations of words, whore-ever tl
accur, tht Header will without trouble underfiand
limfelf } favhtg font* few, which, being of left get
ral Hje> we refer to. he exfUined in their flacts.

•the

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THE FIRST BOOK

• - OF

«

EUCUDE'S

ELEMENTS.

Definitions. :

L A Point is that which has no part.

II. A Line is a longitude with-
out latitude.

III. The ends, or limits, of a
line are Points.

IV. A Right Line is that which lies equally
betwixt its Points.

V. A Superticies is that which has only-
longitude and latitude.

VI. The extremes, or limits, of a Superficies
are lines.

VII. A plain Superficies is that which lies
equally betwixt its lines.

VIM. A piain Angle is the inclination of two
lines the one 10 the other, the one touching the
other in khe lame p»ain, yet not lying in the
fame ftrait line.

IX. And it the lines which contain the Angle
be right lines, ic is cahed a right-lined Angle.

A XWbe*

The firfi Book of

O X. When a right line

CG ftanding upon a right
line AB, makes the angles
on either fide thereof,
CGA, CGB, equal one
% to the other, then both

q "Shofe equal angles are right

angles , and the right
CG, which ftandeth on the other, is
termed a Perpendicular to that (AB) whereon
it ftandeth.

Note, When feveral angles meet at ttyc fame point
{as at G) each particular angle is defcribed by three
letters i whereof the middle letter Jbeweth the angular
foint, and the two other letters the lines that make
that angle : As the angle which the right lines CG,
AG make at G, is called CGA, orAGC.

XI. An obtufe angle is
that which is greater than
a right angle ; as ACD.

AIL An acute angle is
that which is lefs than a
right angle ; as ACE.
TO XIII. A Limit, or Term,
is the end of any thing.

XIV. A Figure is that which is contained
under one or more terms.

XV. A Circle is a plain figure contained under
one line, which is called a Circumference $ unto
which all lines drawn from one point within
the figure, and falling upon the circumference
thereof, are equal the one to the other.

XVI. And that point is
called the Center of the
Circle.

XVIL A Diameter of a
circle is a right line drawn
through the center there-
of, and ending at the cir-
cumference on either fide,

divi-

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Elements

dividing the circle into two equal parts.

XVIlT. A Semicircle is a figure which is
contained under the diameter, and uqder that
part of the circumference which is cut off bj
the diameter.

In the circle EJBCD, E u the center, AC the
diameter y ABC the femicircle.

XIX. Right-lined figures are fuch as are con*
tained tinder right lines.

XX. Three-hded or Trilateral figures are fuch
as are contained under three right lines.

XXI. Fom-fided or Quadrilateral figures are
fuch as are contained under four right lines.

XXII. Many-fided figures are fuen as are con-
tained under more right lines than four.

XXIII. Of Trilateral
figures, that is an Equi-
lateral Triangle, which
hath three equal fides 5
as the Triangle A.

XXIV. Ifofceles is a
Triangle which hath on-
ly two fides equal j ar
toe Triangle B.

XXV. Scalenuxn is a
Triangle whofe three fides
are all unequal; as &

e.

A a

XXVI. Of

The fir[l Book of

XXVI. Of thefe Trila-
teral figures, a right-angled
Triangle is that which has
one right angle , as the
Triangle A.

XXVIL An Amblygo-
nium, or obtufe-angled Tri-
angle, is that which has
one angle obtufe ; as B.

* 9 XXVIII. An 0x7-

gonium, or acute-angled
Triangle, is that which
has three acute angles ;
as C.

An Equiangular, or e-
qual-angled figure is that
whereof all the angles are
equal. Two figures are e-
quiangular,if the feveral angles of the one figure
be equal to the feveral angles of t tjie other. The
fame is to be underftood of Equilateral figures.

lC XXIX. Of (Quadrilate-
ral, or four-fided figures,
a Square js that ^wfiofe fides
are equal, arid angles right j
asABCD. 7

•;■:••/-,

XXX.' A figure on the
one part longer, or a long
fquare, is that which hath
right angles, but not equal
fides j is A^CD.

V

XXXI. A

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EUCLIDE'/ ERmmti:

XXXI. A Rhombus,
or diamond-figure , is
that which has four e-
qual fides , but 13 not
right-angled j as A;

XXXII. A Rhomboid
des, or diamond-like ,fi-
gure, is that whofe op-

!>ofite fides, and oppo-
ite angles, are equal;
but has neither equal
nor right angles 5 as GLMH.

* XXXIII. All other
quadrilateral figures be-
fides thefe are called Tra-
pezia , or Tables : as
GNDH.

J

XXXIV, Parallel, or
equidiftant rightlinesaie
fuch, which being in the
fame fupeificies, if infinitely produced, would
never meet ; as A and B.

XXXV. A Parallelo-
gram is a quadrilateral fi-
gure, whofe oppofite fides
are parallel, or e<
as Gf-HM.

A j XXXVL In

The firfi Book of

XXXVI. In a Paral-
leldgram ABCD, when a
diameter AC, and two
lines EF, HI parallel to
the fides, cutting the dia-
meter in one and the fame
point G, are drawn, fo
that the Parallelogram be
divided by them into four
parallelograms ; thofe two DG, GB, through
which the diameter paffethnot, are called Com-
plements ; and the other two HE, FI, through
which the diameter paiTeth, the Parallelograms
Handing about the diameter.

A Problem w, when fomething is propofed to be
done or effeBed.

A Theoreme is, when fomething is, propofed to le
icmonflrated.

A Corollary is a confe3ary% orfome consequent
truth gained from a preceding demonftratioti*

A Lemma is tfo demonjtration of fome premife9
whereby the proof bf the thing in band becomes the
fiorter. '

Populates or Petitions. "°
I. T7Rom any point to any point to draw a
right line.

2. To produce a right line finite, ftrait fortfc
continually.

j. Upon any center, and at any diftance, to
defcribe a circle.

• Axioms.
*• TPHings equal to the fame third, arealfo
X equal one to the other.

As A -Bin C. Therefore ArrC. Or therefore
all, A, B, C, are equal the one to the other.

Note, When fever al quantities are joined the one
to the other continually with this mark ~ , thefirjh
quantity is by virtue ofthisaxiome equal to the lafi,
and every one to every one : In which cafe we often ab-

flain

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15

EUCEIDE'i metnms:

flam from citing the axiome, for hrcvity's fake \ aU
ibf the force of the confequence depend thereoru

2. If to equal things you add equal things,
the wholes fhall be equal. .

2. If from equal things you takeaway equal
things, the things remaining will be equal.

4. If to unequal things you add equal things,
the wholes will be unequal.

5. If from unequal things you takeaway equal
things, the remainders will be unequal.

6. Things which are double to the fame third,
or to equal things, are equal one to the other.
Underlland the lame of triple, quadruple, &c.

7. Things which are half of one and the fame
thing, or of things equal, are equal the one to
the other. Conceive th$ f*rne of fubtriple,
fubquadruple, &c.

8. Things which agree together, are equal
to the other.

he converfe of this axiome is true in right lines
and angles, but not in figures, unlefs they be like.

Moreover, magnitudes are f aid to agree, when the
parts of the one being apphei to the parts of the other,
they fill up an equal or tie fame place.

9. Every whole is greater than its part.

10. Two right lines cannot have one and the
fame fegment (or part) common to them both.

11. Two right lines meeting in thefame point,
if they be both produced, they fhall neceffarily
cut one the other in that point.

iz. All right Angles are equal the one to the other.

one to
The

ij. UarightlineBAfallingontworigiitlines

A 4 AU, ^o,

Jle firft Book of

AD,CB,tnake the internal angles on the fame fide,
BAD,ABC,lefs than two right angles, thofe two
right lines produced (hall meet on that fide,
where the angles are lefs than two right angles.
14. Two right lines do not contain a fpace,
t$. If to equal things.you add things unequal,
the excefs of the wholes fhall be equal to the
excefs of the additions. ^. ;\ J. 1

16. If to unequal things equal be added, the
excefs of the wholes fhall be equal to the excefs
of thofe which were atfirft.

17. If from equal things, unequal things be
taken away, the excefs of the remainders fhall
be equal to the excefs of the wholes.

18. If'from things unequal, things equal be
taken away, the excefs of the remainders fhall
be equal to the excefs of the Wholes.

19. Every whole is equal to all its parts ta-
ken together.

20. If one whole be double to another, and
that which is taken away from the firft to that
which is taken away from the fecond, the re-
mainder of the fell lhall be double to the re-
mainder of the fecond.

The Citations are to be underflood in this manner \
When you meet with two numbers^ the firft Jbews the
Propofitiony the fecond the Book ; as by 4. 1. you
are to underftand the fourth Propofitiori of the
firft Book ; and fo of the reft. Moreover, ax.
denotes Axiome, foft. Populate, def.Dcfinition,fch.
Scholium, cor. Corollary*

» • » • • •

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EUGLIDE'j foments.

9

PROPOSITION I.

UPon a finite right line

given ABjo defcrihe an
equilateral triangle ACB.

From the centers A and
B,at the diftance of AB, or
BA, a 'crefcribe two circlesa ^fojf.
cutting each other in the
point C ; from whence
h draw two right lines CA,CB. Then is AC cb t.fdft.
= AB<r = BC d— AC. e Wherefore theTri-c i$.def.
angle ACB is equilateral. Winch was to be done, di.ax.

h c aj, def.

Scholium.

m

After the fame manner upon the line ABmay
bedeicribed an Ifofceles triangle, if the diftan-
ces of the equal circles be taken greater or lefs
than the line AB.

PROP. II.

At a foint given A, to make a right line AS
equal to a right line given BC*

From the center jC^it the diftance of CB,* de~a $.poJf.
Tcribe the circle CBff. b join AC ; upon which b r. pojl.
c raife the equilateral triangle ADC. d Produce c 1. 1.
DC to E. From the center D, at the diftance d z.tofi.

of

Digitized by

JO

f l$.def.
gconjlr.
h 3. ax.
k i$.def.
1 1.

Tie jfrj* Bool of

ofc DE, defcribe the circle DEH5 and let DA e
be produced to the point G in the circumfe*-
rende thereof. Then AG ~ CB.

For DG f=. DE, and D A g = DC. Where-
fore AG b = C? k— BC / = Aa WlrtHwas
to he done.

The putting of the point A within or without
the line BC varies the cafes ^ but theeonftru#ioa,
and the demonftration, are every where alike*/

Scbol

The line AG might be taken with a pair of
compaffes 5 but thelo doing anfwers to no Po-
ftulate, as Proclus well intimates*

PROP. III.

Two ijght lines, A and
£Cf Icing given, from the
greater £ C to take away
•the rigj)t line BE equal to
'the lejfer A*

At the point B a draw
the right line BD = A.
The circle defcribed from
the centerB at the Jiftance of BD lhall cut off
b 1 $. del BE irrBD «A.4=BE. Winch wot to le done,
ccmftr. * '

a &• r.

d r. ax.

fPROP.

B d

V* If two triangles BAC% i'JfJF, have two fides of
tie one BA> AC equal to two fides of the other EdL
VF,€acb toits coirefiondent fide (tbatis9BA^ED,

: and

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EUCLIDE'* Elements: x\

and ACz=DF) and have the angle A equal to tie
angle D contained under the equal right lines $ they
JbaU have the bafe EC equal to the bafe EF y and
the triangle BAC JbaU be equal to the triangle EDP*
and the remaining angles &, C, JbaU be equal to the
remaining angles E, r, each to each, under which
the equal fides are fubtended.

If the point D be applied to the point A, and
tlje right line DE plac*d upon the right line AB,
the poinf E fhall fall upon B.becaufe DE a — AB, a hyp.
alfo the right line DFfliall fall upon AC,becaufe
the angle Afl'D.' moreover the point F fliall
fall upon the point C, becaufe AC ar=z DF.b 14. **%
Therefore the right lines EF, BC fliall agree,
becaufe they have the fame Terms, and conse-
quently are equal. Wherefore the triangles
BAC, DEF, and the angles B, E, as alfo the
angles C, F, do agree, and are equal. Which
wot to be Denwnftrated.

PROP. V.

The angles ABC, ACB, at the
bafe of an Ifofceles triangle ABC9
are equal one to the other: And if the
equal fides AB, AC be frodue'd,
the angles CBD, BCE, under the
bafe, JbaU be equal one to the other.

a Take AE^AD $ and * join a j. 1 1
CD, and BE. b ufojl*

_\ Becaule, in the triangles ACD,
■E^ABE, are AB c = AC, and AE cfyp.
&~ AD, and the angle A common to them both, d conftr*
t therefore is the angle ABE - ACD,and the angle e 4. 1,
AEB e — ADC, and the bafe BE e-CD $ alfo
EC / ss. DB. Therefore in the triangles BEC, f 2. ax.
BDCg fhall be the angle ECB - DBC. Which g 4. 1.
was to be Dem. Alfo therefore the angle EBC^
DCB. but the angle ABE £=ACD 5 therefore h before.
the angle ABC AsACB. Which vat to be Dem. k 3. ax.

CoroU.

■

■

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iz Tie firfi Bool of

Coroll.

» Hence, every equilateral triangle is alfo
equiangular.

PROP. VI.

If two angles ABC, ACB of a
triangle ABC be equal the one to
the other, the fides AC, AB fubtended
under the equal angles, Jball alfo
he equal one to the other.

If the fides be not equal, let one
be bigger than the other, fuppofe BA c~ CA. a
Make BDrzCA, ami b craw the line CD,
b i. poft. In thc triangles DEC, ACB, becaufe BD cr=z
cfuppof. £^ and the fide BC is common, and the angle
DBC d - ACB, the triangles DBC, ACB e {hall
be equal the one to the other, a part to the
whole. fmkbuimpoMlc.

Coroll.

Hence, Every equiangular triangle is alfo
equilateral.

PROP. VII.

% J. I.
I. pt
fupp
d hyp.
t 4, 1.

f p. ax.

Upon the fame right line AB two right lines being
irawn AC, BC, two other right I hies equal to the
former , AD, BD, each to each (viz. AD — ; AC9
and BD — BC) cannot be drawn from the fame
points J, B, on the fame fide C, to fever al points, as
C and D, but only to C.

1. Cafe. If the point D be fet in the line AC,
a p. ax. it is plain that AD is a not equal to AC.

z. Cafe. If the point D be placed within the
triangle ACB,then draw the line CD,and produce
BDF, and BCE» Now you would have AD^ AC.

then

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EUCLIDE'* Elements. x%

then the angle ADC JrrACD; as alfo, becaufe b <. i.
BD cr= BC, the angle FDC r=ib ACD. there- c fuvpoL
fore is the arlgle FDC cr d ACD, that is, the d Q.ax.
angle FDC c" A DC. d Winch is impoJJibU.

$. C/i/e. If D falls without the triangle ACB,
let CD be joined.

Again the angle ACD e— ADC, and the e <.*.
angle BCD e = BDC. / Therefore the angle f o a*.
ACD cr BDC, viz. the angle ADC <r BDC. *
JTfcVi if imfojjihle. Therefore, fire

PROP. VIII.

If two triangles
ABC, DEF, have
two fides A By AC
equal to two fides.
X>£, D F, *j<rfi to
c^7; , and the hafc
BC eaual to the hafe £F, then the angles con-
tained under the equal right lines JbalL he equal,
viz. A to D.

Becaufe BC *—EF, if the bafe BC be laid on a hyp.
the bafe EF,ithey will agree: therefore whereas b ax. 8.
AB c—DEj and AC—DF, the point A will fail c hyp.
on D (for it cannot fall on any other point, by
the precedent proportion) and fo the fides of
the angles A and D are coincident ; d wherefore d 8. ax%
thofe angles are equal. Which wot to he Dem.

CoroU. *•

4

I. Hence, Triangles mutually equilateral, are
alfo mutually c equiangular. e 4. 1

a. Triangles mutually equilateral, $ are equal ** %

one to the other.

PROP,

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a J. i.
b i. im

14 The firfi Book of

PROP. IX,

To bifeS9 or divide into two
equal parts, a right-lined an-
gle given B AC.

a Take AD =r to AE,
and draw the line DE ; up-
on which b make an equi-
lateral triangle DFE. draw
the right line AFj it fliall
bifeft the angle,
c conftr . For AD <r=AE> and the fide AF is common,
d & u and the bafe DF<?— FE. d therefore the angle
DAF = EAF. Which was to be done.

Coroll.

Hence it appears, how an -angle may be cut
into any equal parts, as ^, 8, 16, Sec. to wit, by;
bifefting each part again.

The method of cutting angles into any equal
parts required, by a Rule and Compafs, is as
yet unknown to Geometricians.

PROP. X.

To bifeS a right line given

Upon the line given AB
a ere. ft an equilateral triangle
ABC; andibifeftthe angle C
with the right line CD.
That line fliall alfo bifeft the
line given AB.
ceonftu For ACfrrBU, and the fide CD is common,
d 4. 1, and the angle ACDc— BCD. therefore AD^BD*
Which wot to be done.

The praftice of this and the precedent Propo-
fition is eafily fhewn by the conftruftion of the
x Prop, of tlus Book.

PROP,

a tm u

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EUCLIDE7 Elements] if

PROP. XL

From a point C in a
right line given IB to e-
rett a right line CF at
right angles.

a Take on either fide
of the point given CDa * u
r=z C E. upon the right

E B line DE b ereft an equi-5 X- 1#
lateral triangle, draw the line FC, and it will
be the perpendicular required.

For the triangles DFC, EFC are mutually <re-c confix.
quilateral ; d therefore the angle DCFr^ECF. ed 8. U
therefore FC is perpendicular. Wlrieh was to be done, e 10. 1
7he prafticeof this and the following is ea-
performed by the help of a fquare.
PROPi Xll.

Upon an infinite
tight line given JBf
rfrom a point given
that is not in it, to let
_ fall a perpendicular

^^M^^ From the center

C a defcribea circle cutting the right line given
AB in the points E and F. Then bifeft EF in a 5. pofi.
G, and draw the right line CG, which will be b 10. i%
the perpendicular required.

Let the lines CE,CF be drawn. The triangles
EGC, FGC are mutually c equilateral, d there- c confir.
fore the angles EGC, FGC are equal, and by d 8. u
c confluence right, e Wherefore GC is a per- e io.<tef.

pendicular. Winch was to be done.

PROP. XIII.

Wlien a right line AB fianding
upon a right line CD maketh angles
JJIC9 ABD 5 it maketh eitfor two

right angles^ or two angle; equal to

•Dtworight.

«»

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j6 The firjl Book of

a def. io. If the angles ABC, A Bp be equal, a then they
make two right angles } if unequal, then from the
b it. i. P°in* B- b let there be erected a perpendicular BE.
c 19. ax. Becaufe the angle ABC c = to a right -1- ABE,
d x.ax. and the angles ABD d — to a right — ABE,
« 2. therefore fhall be ABC ABD e rr to two
right angles -4- ABE — ABE — two right an-
gles. Which vp<v to he demonftrated.

Corollaries.

1. Hence, if one angle A B D be right, the
other ABC is alfo right; if one acute, the
other is obtute, and fo on the contrary.

2. If more right lines than one Hand upon the
fame right line at the fame point, the angleg
ihall be equal to two right.

3. Two right lines cutting each other m^e
angles equal to four right. k

4. All the angles m^de about one point, mala
four right $ as appeajk by Coroll.A. 1

P R ^R. XIV.

If to any right line AB, and a
joint tberXjn B, two right lines, not
•m drawn fronl the fame Jide, do make
~ the angles ABC, 4fiD on each fide
T equal to two right, the'lines CByBD
fball make one ftrait line. i ,
If you deny it, let CB, BE make one right line,
a \%. I. then ihall be the angle ABC ABE arrtwo right
b hyp. angles b = ABC -t- ABD. Which is abfurd.
c 9. ax.

PROP. XV.

If two right lines AB, CD cut

thro? one another, then are the two

angles which are ofpotiie, viz. C£JS9

TTAtD, equal one to the other.

For the. angle A E C -+ C E B

a — to two light angles err AEC

^AED; i therefore CEB=ALD. Which was
f>l*ax. to h demonftrated^

Schol

1

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EUCLIDE', Elements.

Sthol

*7

If to any right line GH, and in it a point A,
two right lines being drawn EA, AF, and not
taken on the fame hde,make the vertical(oroppo-
fite) angles IXandB equal, thofe right lines EA,
AF, do meet diredly and make one ftrait line.

For two right angles are a equal to the angle * T*
D-^AtfrB \ A. b therefore, I A, AF, are in a b x4- *•
iirait line. Which was to be demonjlrated.

V* * If fou^^ht lines EA, EB, EC,
A \m ED, proceeding from one point
"T y E, make the angles verucally
\ oppofite equal the one to the
\D other, each two lines, A fc, EB,
and CE, ED, are placed in one ftrait line.

For becaufe the angle ALC < A ED |- CEB
+ DEB a<z. to 4 right angles,therei<jre the angle „ M „ ... * •
AEfi -h AED> = CElS t DEB =r to wu£4£f£l
right angles, c therefore CED and AEti aie,
ftrait line*. JTZ^ J was to be demonjlrated. 1 a x *

c 14. r.

PROP. XVI.
p One jide E C of any trian-
gle ABC being produced, the
outward angle ACD will be
greater th>n either of the in-
(t_ ward and opfujite angles C A By

p i>CB A.

Let the right lines AH, EE
a bifeil the tides A C, BC;aio.i.&
b produce EF =; BE, and Hl,i. poft.

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i8 The firfi Book of

I — AH. and join FC, and IC ; and produce
ACG.

c conjlr. Becaufe CE c^HK, and EF c= EB, and the
d 15. i. angle FEC d-zz BEA, the angle ECF e fhall be
c 4. 1. equal to EAB. By the like argument is the an*
f 15. x, gle ICH ~~ ABH. Therefore the whole aogle
g 9, ax. ACD (7"BCG)£is greater than either the angle
CAB or ABC. Which was to be dcmonftrated.

PROP. XVIL

/

Two angles of any triangle
JBC, which way foever they
he taken, are lefs than two
right angles.
Let the fide B C be pro-
C Pduced. Becaufe the angle
ACD -|- ACB a—z right
a 15. 1. angles, and theanglfpACD Izr A, c therefore
o i<5. 1. A-j ACB-3 then two right angles. Afterthe fame
c 4. ax. manner is the angle B -\- ACB T3 then two
right. Laftly, the fide AB being produced, the
angle A 1 B will be alfo lefs than two right an-
gles. Which was to he demonjtrated.

CorolL

1. Hence it follows that in every triangle,
wherein one angle is either right or obtufe, the
vtwo others are acute angles.

2. If a right line AE make unequal angles
with another right line D, one acute AED, the
other obtufe AEC, a perpendicular AD let fall

£ •£ j) from any point A to the other line CD, lhall
fall on that fide the acute is of.

For if AC, drawn on the fide of the obtufe an-
gle, be a perpendicular, then in the triangle A£C
T7* *' mall AEC j ACE be* greater than two right an-
gles. Which is contrary to the prectdent Ptof.

3. AH the angles of an equilateral triangle,
and the two angles of an llofceles triangle that
aje up on the bale, are acute.

PROP.

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EUCLIDE'i Element^ 19
PROP. XVIII

The greateft fide AC of every
triangle ABC fub tends the great-
ft angle ABC. , .

From AC a tike away ADa 3. 1.
'AB, and join BD. h There- b $. 1.
^ir ^^fore is the angle ADBrr ABI).

But ADB f ir C ; therefore is ABD cr C ; d there- c To-# t.
fore the whole angle ABC cr C. After the fame d 0, ax.
manner, lhall be ABC c~ A. Whichwas to he dem.

PROP. XIX.

In every triangle ABC, under >
the greateft angle A is fubtended

■

"or if AB be fuppofed equal
BC, then will be the angle A
'a r= C, which is contrary to the
Hypothefis : and if ABcrBC, then fhall be the an- * >' *"
gle C be A, which is againft the Hypothelis. t q
Wherefore rather BC cr AB 5 and after the fame
manner BC cr AC. Which was to be dem.

PROP. XX.

Of every triangle ABC two
fides BAy AC, any way taken, are
greater than the fide that remains
BC. r*
Produce the line B A, a and a }• u
Ctake AD — AC, ana draw the
line DC, b then fhall the angle D be equal to b 5. U
ACD, c therefore is the whole angle BCDr*D; c 9. ape.
d therefore BD (e BA \ AC) cr BC. Which was d 19. 1.
to be demonftrated. e conjlr.

PROP. XXI. &i.ax.

If from the utmoft pint sdj ''one
fide BC of a triangle A B C two
right lines B D, CD be hawn
to any foint within the triangle,
then are both thofe two lines
Jborier than the two other fides of
>C B z ^ the

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20

a 2o. i.
b 4. ax.
c 16. 1.

rtbe fir (I Book of

the triangle^ BA, CJ ; but do contain a greater an*
gle, £D(J.

Let BD be produced to E. Then is CE -+ ID
ar-CD,,and BD common to both, b then fhallbe
BD DE+ECc-CD -BD. Again, BA AE
act BE. b therefore BA AC -BE 1 EC. Where-
fore r. BA l-AC c-BD *DC.i. The angle BDC
c cr DEC c cr A. Therefore the angle BDCirA.
Which was tobe demonftrated.

PROP. XXII.

Jl B C

a j. i*

To make a triangle FKG of three right lines FK9
FG9 GK9 which jball be equal to thee right lines
given A9 B9 C. Of which it is neceffary that any
two taken together be longer than the third.

From the infinite line DE a take DF, FG,GH
equal to the lines given, A, B, C. Then it from
the,* centers F and G by the diftances of FD
D I'fojt. and GH, two circles be drawn cutting each
* other in K, and the right lines KF, KG be join-
c 1 5- def. ej, the triangle FKG fhall be made, c whofe
fides FK, FG, GK are equal to the three lines
DF, FG, GH d that is to the three lines given
A, B, C. Wliich was to be done.

PROP. XXIII.

d 1. ax.

a 1. poft.

At a pint A in a right
line given AB to make a
right-lined angle A equal
to a right-lined angle
given D. ,

Draw the right
* line

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EUCLIFE'x Elements. 21

line CF cutting the fides of the angle given any
ways; imake AG~CD;upon AG %£raifeatrian- b 3. t.
gle equilateral to the former CDF, lb that AH be c 21. I.
equal to DF, and Gh to CF. th:n fhall you have
the angle AJ^D.* Which was to he done. d 8. 1.

PRO P. XXIV.

If two triangles ABC, DEF have two fides of
the one mangle AB, AC equal to two fides of the
other triangle DEyDFy each to other , and have the
angle Ajrreater than the angle EDF contained ww- ^
der the/equal right lines , they Jball alfo have the
hafe EC greater than the hafe EF. t

a Let the angle EDG bec5*d£de equal to A, a *•
and the fide DG h m DF c d^AC ; and let EG, b 1.
and FG be joined. ^ c typ.

1. Cafe. It EG fall above EF ; becaufeAErf— d hyp.
DE, and AC* -DG, and the angle A e~EDG, e conjtr.
f therefore is BC- EG. But becaufe DF DG, f 4- *•
^therefore is the angle DFG^DGF; ^therefore g 5- u
is the angle DFG - FFG, and by eonfequence h 9. ax.
the angle EFG h cr EGF, k wherefore EG <BC) k 10. 1.
c-EF.

B j z.Cafe.

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?

i

is-

r

22

73* j&y? */

into

2. Cd/e. If the bafe EF falls in ihe fam* place
1 9- dAT. with EG, / it is evident that EG (BC) cr EF.

J. Cafe. If EG fall below EF, then becaufe DG
m2M, -\- GE wrDF + FE, if from bothJ)a,DFbe
taken away, which are equal, EG (BC) remains
n J. d*. 7i cr EF. JP£k& mi /a be demonjlrated.

a 4. 1.

-

1)24.1.

-

■ ■

PROP. XXV.

Jf /wq triangks ABC*
DEF have two fdesAB,
AC eaual to two fides
DE, vF, each to, other*
sind have the bafe BC
: greater than the bafe EF9
they fb all alfo have the angle J contained under the
equal right lines greater than the angle A

For if the angle A be faid to be equal to D, a
then is the bafe BC = EF, which is againft the
Hypothecs. If it be faid the angle A I>,then b
-will be BC -a EF, which is alfo againftthc Hyp.
Therefore BC cr EF. Which was to h dem.

' PROP. XXVI.

If two triangles BAC, EDG have two angles of the
one £>C equal to the two angles of the other £, DGE9
each to his Cf/rrejjtonient angle^ and have alfo one fide
of the one equal to one fide of the other ^either that fide
which lyeth betwixt the equal angl&s^or that which is
fubtendedunderoncofthe equal angles j the other fides

alfo

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EUCLIDE'j Elements. 23

alfo of the ant flail be equal to the other fides of the
other each to his correflondent fide, and the other
angle of the one flail be equal to the other angle of

the other. , . . ,

1. Hyp. Let BC be equal to EG, which are the
fides that lie between the equal angles. Then I
fay BA— ED, and ACrrDG, and the angle A=
EDG.ForifitbefaidthatEDcrBAjthenaletEHa u

be madeequal to BA,and let the lineGHbe drawn.

Becaufe AB *=HE, and BC c ~ EG, and th^MM:
angle B f —E, therefore (hall be the angle EGH <j tyf.
d^ Ce^z DGE. / Which is ahfurd. After the d 4. I-
fame manner let AC be equal to DG, d then will e ■
the angle A be equal to EDG. * * ax*

2. Hyp. Let AB be equal to ED. Then I fay
BCrrEG,and AC=DG,and the angle A^EDG.
For if EG be greater than BC, make El=BC, *nd
join the line T)L Now becaufe AB*=ED, andg

BC h = EI, and the angle G^ = E^ therefore h /^7-
will be the angle EIDfe-C/^EGD. m Wbkb^Ar^
ixaJ/ari. Therefore is BC=f G, and fo as ; be-|J W-
fore AC-DG, and the angle A=EDG. PMci* 16. !•
*>d ; to demonftrated.

PROP. XXVII.

a rigfo I™ falling
^ two r%ht lines JP, CD,

7 ^Swrt/tf /ie alternate angles

/JF T> DFE, equal the one

to the other, then are the right lines AB, CD parallel.

If AB CD be faid not to be parallel,produce
them till'they meet in G. which being fuppofcd,
the outward angle AEF will be a greater than a itf. ^
the inward angle DFE, to which it was equal
bv Hvoothefis. Which is repugnant.

PROP. XXVIII.
m If a right line E F falling
jk / upon two right lines, J B, CD

J5 mafa ij}C outward angle AGE

C At m P of the one line equal to C HG
-/a the inward and oppose angle
\ B 4 of

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24 The firjl Book of

of the other on the fame fide, or make the inward an-
gles on the fame fide AGH, CHG equalto two right
angle sy then are the right Iwes AB, CD parallel.
Hyp. i. Becaufc by Hypothefis the angle AGE
a H. I. =r CHG, a therefore are BGH, CHG alternate
b 27. 1. angles and equal : And h therefore are AB and
CD parallel.
Hyp. z. Becaufe by Hypothefis the angle AGH
* t*. I* -4. CHG ■=■ to two right, a — AGH - BGH,
b J. ax. h thence is CHG — BGH ; and c therefore AB,
C 27. U CD are parallel. Which was to be demonftrated.

PROP. XXIX.

If a right line EF fall upon two
parallels, AB, CD, it will both
make the alternate angles DHG,
AGH equal each to other, and
the outward angle EGE equal to
the inward and opfofite angle 071
the fame fide DHE* as alfo the inward angles on the
Jame fide AGH, CHG equalto two right angles.
It is evident that AGH 1 CHG r. 1 right an-
ai;. ax. gles^ a othei wife AB,CD would not be parallel,
b *
e
d

5

a zo. r.
b j. ax.

Coroll,

Hence it follows
that every paralle-
logram AC having
one angle right A,
the reft aie * alfo
Bright.

For A - B a =r z right angles. Therefore
whereas A is right, b muft B be alfo right. By.
tfre fame argument a*e C and D right angles.

?KOP,

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EUCLTDEfx Elements.

PROP. XXX.

i??^/^ lines (AB, CD) parallel
.JJ to one tf?2<4 r/;e /awe right line,
EF\ are alfo parallel the one to
* the other.

-JD Let GI cut the three right
lines given any ways, then
becaufe AB, EF are parallel, will be the angle
AG I a — EHI. alfo becaufe CD and EF are pa-a 2p#I#
raltel, will be the angle EHI a— DIG. h The.e- 5 l^ *a#9
fo»e the angle AGI — DIG. c whence AB andc Zj% ii
CD are parallel. Which was to he dem. i

PROP. XXXI.

a From a point given A to draw a

E / ' ^ 'right line AE paiallel to a right
J line given BC.

A'

J)

C From the point A draw a right
line AD to any point of the given

right line ; with which at the point thereof
a A make an angle DAE - ADCithen will AE a 2J. I.
and BC be parallel. Which was to he done. b zj. r.

R R O P XXXIL

Of any triangle ABConeJide
]3J £C, heing drawn out, the out-
ward angle ACD JbaU he equal
to the two inward oppufue an-
gles Ay B9 and the tmee inward
j^angles of the triangle, A, B%
ACB, Jhall he equal to two right angles.

From C a draw CE parallel to BA. Then is a ji. 1.
the angle A h- ACE, and the angle B h LCD. b zo. r.
Therefore A * BcACE \ ECD d = ACD. c 1. ax.
Which was to he dtmonftrated. s * 19- «*•

I affirm ACD \ ACB e = two right angles ; e 1.
/ therefore A-tB-f ACB= 2 right angles. Which i 1. ax.
was to he dcmovjlrated,

Coroll.

1. tjiree angles of any triangle taken toge-

r

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The firjt Book of

ther are equal to three angles of any other trian-
gle taken together. From whence it follows,

2. That if in one triangle, two angles (taken
feverally, or together) be equal to two angles of
another triangle (taken feverally, or together)
then is the remaining angle of the one equal to
the remaining angle of the other. In like man-
ner, if two triangles have one angle of the one
equal to one of the other, then is the fumof the
remaining angles of the one triangle equal to the
fum of the remaining angles of the other.

3. If one angle in a triangle be right, the
other two are equal to aright. Likewife, that
angle in a triangle which is equal to the other
two, is it felf a right angle.

4. When in an Ifofceles the angle made by the
equal fides is right, the other two upon the bafe
are each of them half a right angle.

5. An angle of an equilateral triangle makes
two third parts of a right angle. For { of two
right angles is equal to ? of one.

ScboL

By the help bf this Proportion you may know
how many right angles the inward and outward
angles ot a right-lined figure make j as may ap-
pear by thefe two following Theorems.

THEOREM I.

Jll the angles of a right-lined figure do together
make twice as many tight angles^ bating fourt as
there are fides of the figure.

From any point within the figure, let right

lines

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EUCLIDE'j Elements. 27

lines be drawn all thro* the angles of the figure,
which ftiall refolve the figure into as many tri-
angles as there are fides or the figure. Where-
fore, whereas every triangle affords two right
angles, all the triangles taken together will
make up twice as many right angles as there
are fides. But the angles about the faid point
within the figure make up four right -> there-
fore, if from the angles of all the triangles you
take away all the angles which are about the
faid point, the remaining angles, which make
up the angles of the figure, will make twice as
many right angles, bating four, as there are fides
of the figure. Which was to be denu

CoroU.

Hence, All right-lined figures oPthe fame
fpecies have the lums of their angles equal.

THEOREM if. .

Jll tlx outward angles of any right-lined figure^
taken together, make up four right angles.

For all the. feveral inward angles of a figure
with the feveral outward angles of the famemake
two right angles s therefore all the inwaid angles,
together with the outward, make twice as many
right angles as there are fides of the figure; but ,
(as itVas nowihewn) all the inward angles with
four right, make twice as many right as there are
fides of the figure ; therefore the outward angles
are equal to four right angles. Which was to bedem.

CoroU

All right-lined figures of whatfoever fpecies
have the Turns of their outward angles equal.

PROP. XXX1IL

If two equal and parallel lines
JB9 CD be joined together with
two other right lines JC, BD, then

C ^ *r are thofe lines alfo equal and parallel.

Draw a line from C to B. NowbecauleAB and
CD are paiallel, and the angle ABC a = BCD ; a 29. r.
and alfo by Hypolhcfi»AR=:CD,«id the fide CB

com- .

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28 The firft Bool of

b 4. 1. common, therefore is AC b ttBD, and the an*
C27. 1. gle ACB b = BD . f whence alfo AC, BD
are parallel.

PROP. XXXIV,
Lg 7n parallelograms, as ABCD,
the oppofi(€ jides AByCD, and

Vr*^ ED, are equal each to other z

^ " &and the oppofite angles J, D, and

ABD^ACD are aljo equal ; and the diameter EC b'u
fefts the J ante.

a hyp* Became A B, CD a are parallel, b therefore is
the angle ABC - BCD. Alfo becaufe AC, BD

b 19. 1. a<e a parallel, b therefore is the angle ACB rr

c z. ax% CBD ; c therefore t he whole angle ACD -r ABD.

Arttr the fame manner i^ A D. Moreover be-
caule the angles ABC, ACB lie at eacn end of the
fide CB, and are e^ual to BCD,CBD. i therefore

d 16. 1 * is AC - BD, and AB d CD, and fo the trian-
gle ABC ~ CBD. Which was to be dm.

SchoL

Every four fid ed figure ABDC having the oppofite
fides equal, is a parallelogram.
a 17. 1. For by 8. 1. the angle ABC - BCD ; a where.

fore AB, CD a<e parallel. In like manner is the*
b j $. def. angle BC A - CBD ; a wherefore AC, BD are
l. alio parallel, b Therefore ABCD is a parallelo-

gram. Winch was to be demonfi rated.

32 \ From hence

A. 1 )p B may be learned

how to draw a
parallel CD to a

right line given
AB, thro' the point affigned C.

Take in the line AB any point, as E. From
the centers E and C at any diftance draw two
equal circles EF, CD. From the center F by the
fpace of EC d*aw a circle FD, which fhall cut the
former circle CD in the point D. Then fhall
the line drawn CD be parallel to AB. for as it was
before demonftrated, CEFD is a parallelogram.

PROP.

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EUCLIDF* Elements.

*

PROP. XXXV.

Parallelogramt, BCDJ9
BCEF, which ftand upon
the fame bafe EC, and
between the fame paral-
lels JF, EC, are equal
one to the other.

B C For AD j r- EC jrra J4.T.

EF, add DE common to both, b then is AE ~ 5 z. a9Cm
DF. But alfo AB a — DC, and the angle Ac— c 19. i.
CDF. d Therefore is the triangle ABE DCF. d 4. 1.
take away DGE common to both triangles, ee 3. ax*
then is the Trapezium ABGDrz: EGCF. add
BGC common to Doth, /then is the patailelo- f 2, ax.
gram ABCD EBCF. Which was to be dan.

The demonftration of any other cafes is not •
unlike, but much more plain and eafy.

SchoL

If the fide AB of a right-
angled parallelogram ABCD be
conceived to be carried along
perpendicularly thro' the whole
line BC, or BC fliro' the whole
line AB, the Area or content of %
the Redangle ABCD fliallvbe
produced by that motion. Hence

A

!

12

by the drawing or multiplication of two conti-
guous fides. For example's lake \ let AB be
luppofed four foot, and BC be three : draw 5
into 4, there will be produced 11 fquare feet
for the Aiea of the Redtangle.

This being luppofed, the dimenfion of any pa-
rallelogram (*EBCF) is found out by this Theo- * $ee the
rem. *or the Area thereof is produced from the figure of
altitude BA dtawn into the bafe BC. SotheAiea Prop. }$•
of the parallelogram AC = EBCF, is made by
the drawing of JtJA into BC, therefore, &c

PiiOP.

Digitized by

jo The firfi Book of

PROP. XXXVI.

A !D E *P Parallelograms, BC-'

I ^\ DJ> GHFE>ftand-

/ -^/O^r^^ \ 2W£ vPon e9ual bafes

fe^— X 1 * B C\ G H% and betwixt

B C Q H /in/i* farattels AF>
B H, are equal to the other.
a fyf. Draw BE and CE. Becaule BC a = GH b =
b 34. 1. EF, <: therefore is BCEFaparallelogram.Whence
c 33. i. * the parallelogram BCD A d rrBCFE d— GHFE.
d 55. 1. Winch was to be dem.

XXXVII.

Triangles , iCi/,
BCD, ftanding upon
the fame bafe BC, and.
between the fame parol-

Ids BC, EF, are equal

E C one to the other.

a 31. 1. * Draw BE parallel to CA, a and CF parallel
b 34. 1. to BD. Then is the triangle BCA b r=r ± of the
c 3 %.\.and parallelogram BCAE c = i. BDFC i = BCD.
7.d#. JPfoVfi nwto be dem.

PROP. XXXVIII.

jy w Triangles, BCA, EFD,
rfet upon equal bafes BC,EF,
and between the fame pa-
rallels GH, BF, are equal
the one to the other.

Draw BG parallel to
CAand FM parallel to
ED. Then is the trian-

a 34- 1. Sle BCA a = T Pgr- BCAG b = a EDHF c

b 36.1.004 EFD. A^JfrA was to be dem*

7*dx. x SchoL

c 34. 1. If the bafe BC be greater than EF, then is
the triangle BAC cr EDF, and fo on the con-
trary.

PROP.

Digitized by

EUCLIDE'j Elemnti.

PROP. XXXIX.

fjp Equal triangles BCA9
BCD,, ftanding on the
fame baje B C, and on
the fame fide, they are
alfo between the fame
parallels AD, BC.
If you deny it, let another line AF be parallel
to BC 5 and let CF be drawn. Then is the trian-
gle CBF jitCBA *~CBD. c lVlnch is abfurd. a ^7. r.

b hyp.

PROP. XL- c?.ax.

Equal triangles
BCA, EFD, Jlandvng
upon equal bafes BL\
EF, and on the fame
fide, they are betwixt »
the fame parallels.
If you deny it, let
another line AH be parallel to BF, and let FH
be drawn. Then is the triangle EFH a =BCA a 58. 1.
I ss EFD. c Which is abfurd. b hyp. s

c 9. ax*

PROP. XLI.

» 3 If * Pgr- ABCD have the .

1 fame bafe BCwith the triangle

BCE, and be between the fame ■ 1
parallels AE, BC then is the
Pgr. ABCD double to the
triangle BCE,
Let the line AC be drawn. Then is the triangle
BCA a = BCE- therefore is the Pgr. ABCD. b a j7. u
— z BCA c = 2 BCE. Winch was to be dem. b 54. I.

Schol. c 6. ax.

From hence may the Area of any triangle BCE
be found, for whereas the Area of the Pgr.ABCD
is produced by the altitude drawn into the bafe,
therefore fhallthe Area of a triangle be produced
by half the altitude drawn into the bafe, or half

the

/

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3*

a ;r. i.
b i], u
? 10. i.

a 3i

c4i

i.
. x.

firfi Book of
the bafe drawn into the altitude, as if fo be the
bafe BC be 8, and the altitude 7, then is the
Area of the triangle BCE, 28.

PROP. XLII.

To make a Pgr. ECGF equal to a triangle given
ABC maii angle equal to a right- lined angle given A.

Through A a draw AG parallel to BC b make
the angle BCG ~ B. c bifeft the bafe BC in E,
and draw EF parallel to CG. then is the pro-
blem refolved.

PROP. XLIII.

a 34. 1.

b j. ax.

In every Pgr. AB CD9
the complements DG, GS
ofthofe Pgrs HE, F J,
which Jl and about the dia-
meter, are equal one to the
other.

For the triangle ACD
a = ACB, and the triangle AGH.1 AGE,and
the triangle GCF a GCI. b Thoetore the
Pgr. DG a BG. Which was to be dem.

PROP.

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EVCUDE'/ Wi.

33

0

f rop, nwifj

. . * . . •

• !

I

D S

-A» KM

Z7f ott rigfo /iwe ro Tftafce 5 Parallels

gram FL at a right-lined angle given C, equal to A
triangle given B. ' f

a Make a Pgr. FD r= to the triangle B, fo a 42; I i
that the angle GFE may be equal to C; Pro-
duce GF till FH be equal to the line given A.
thro* H b draw IL parallel to EF, which let fc ji# t.
DE produced meet in the point I. let DGL s
drawn forth meet with a right line drawn from
I in the point K. thro' K b draw K L parallel
to GH, with which let EF drawn out meet
at M, and IB at L. Then fliall FL be the Pgr.
tequired.

tot the Pgr. FL c £= FD = B, rfand the angle C 43* ft
MFH=.OFE=C. Which wu to be done. d 1 5« *Z

PROP* XLVa -i,

. *1«X HI***!**. '

e a hi ;
I — f-f

'B T / .

f f i

OjpOB a yi^Jf line given FG to make a Pgr. FL
Bqual to a right-lined figure given J£CD, at *
right-lined angle given £•

.1

r

4.X

* M "X

/

oogle

■

34

The firfi Book of

Refolve the right-lined figure given intotri-

ZD.

* 44. 1. angles BAD, BCD. then a make a Pgr. FH =
BAD, fo that the angle F may be equal to E.
FI being produced, a make on HI the Pgr. IL=

b 19. ax* BCD. Then is the Pgr. FL b = FH IL c =

c covftr. ABCD. Which was to be done.

Schol.

H 2*

□

Hence is eafily found the excefs,HE,vrhereby
any right-lined figure, A, exceeds a lefs right-
lined hgure,B; nameiyjf to fome right Hne,CD,
both be applied, Pgr. DF^iA, and DH~B.

PROP. XLVI.

Upon a* right' lint
riven At) t* defcrih a
fqvaie AC.

a Ereft two perpendi-
culars AB, DC, b equal
to the line given AD ;
then join B C, and the
D thing required is done,
c corfr. For, whereas the angle A^-Dcm right, d
d 18. 1. therefore are AB, DC parallel. But they are alfo
ft conjtr. t equal \ / therefore AJD, BC are both parallel
f 34. r. and equal ; therefore the figure AC is a Pgr.
g/<5.2o.T« and equilateral. Moreover the angles are all
h 29. def. right, g becaufe one, A, is right 5 h therefore
AC is a f(JUare- IVhhch was to be done.

After the fame manner you may eafily . de-
fer ibe a Re&aflgle contained under two right
lines giveto

m

*;- PROP.

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EUCLXDE's Ekmmts.

IS

PROP. XLVII.

in right an-
gled Ttiangles j
£ A C, the fquare
£Ey which is made
of the fide BC that
Ifubtends. the right
angle £ A C is
equal to both the
fquares BG, CHf
which are made
of the fides, J£9
AC containing the
right angle.

{oin AE, and
); and'diatf

J> MB

AM parallel to CE.

Becaufe the angle DBC a = FBA, add the a ii. ami
angle ABC common to them both ; then is the t
angle ABD — FRC. Moreover AB i^rFB, and
BD b-r BC ; c therefor is the triangle ABD b *p. ief*
FBC. But the Pgr. BM ^ z ABD, and the c 4. 1.
Pgr. d BG 2 z FBC (for GAG is one fight lined 41. u
by Hyp. and 14. r.) e therefore is the Pgr.4$M= e 6. ax,
BG. By the fame way of argument is the P
CM = CH. Therefore is the whole BE=r
CH. Winch was to be d$m.

1 * • > • •..«».•
• . • * \ . '

Schoh

^. • s ■

This moft excellent and ufeful Theorem has
deferved the title of Pythagoras's Theorem, be-
caufe he was the In venter of it- By the help of
which the addition and fubftiaftion of fquares
are performed ; to which purpofe ferve the two
following Problems.

■ » 1

Jn3r.
iTacg.

a if. r.

M7* *•

The firfi Book of

■

PROBLEM I.

To wafe* oTjf fquare equal
to any number of J quarts K
given. /
< Let thf ee fqUares be given
jr whereof theftdesareAB,BG,
* CE. a Make the right angle
FBZ having the fides infi-
nite; and on them transfer
BA and BC; join AC. then
is ACq b == ABq BCq.
* Then transfer AC from B
^ to X, and CE the third fide
B given from B to E; ioin
EX. * Then is EXq = EBq ( CEq ) -+ BXq
€ 2. 4** (ACq) * — CEq + ABq B(iq. Which was ft

PROBLEM II.

7W unequal right lines
being given At, 2?C, to make
a fquare equal to the diffe-
rence of the two f quarts of the
iven lines, JB, EC*
From the center B, at the
liftance of BA, defcribe a circle s and from the
point C ereft a perpendicular CE meeting with
ft 4?. l« the circumference in E ; and draw BE. a Then
\>l.ax. is BEq (BAqV= BCq + CEq* b Therefore SAq
f BC<j = CEq. Which was to be done.

■4 V

4\ vt

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Googl

EUCLIDE'x Elementi.
PROBLEM III.

37

CJ Any two fides of a right-angled
triangle ABC being known , to
find out the third.
/ Let the fides AB, AG en-

/ compafiing the right angle, be,
' c the one 6 foot, the other 8.
s Therefore, whereas ACq+ABq
= 64 36 loo = BCq,

thence is BC = V 100 = IO« 47-
Otherwife, let the fides AB,
j BC be known, the one fix foot,
_ A the other 10, Therefore fince

BCq — ABq =100 — 36 r=64^ACq, thence 47'
is AC = V ^4 = 8. 7^2>ic J w^f jo £e <*Vw.

PROP. XtVIII.

1 -\ frit. ^ > T.; 'f.l Jg

|H Jjf uir* m upon one fide
BC of a triangle be equal to the
fquares made on the other fides
of tfo triangle AB, AC, then
the angle BAC comprehended nn-
C der thofe two other fides of the
triangle AB, AC, is a right angle*

Draw to the point A in AC a perpendicular
line DA— AB, and join CD.

Now is a CDq-ADq |- ACq^ABq+ACq a 47. tl
rr:BCq. * Therefore is CD— BC, And therefore * Sec the
the triangles CAB, CAD are mutually equila- Mowing
teral. Wherefore the angle CAB Jr^ CAD c Theor.
5= a right angle. Winch was to he dent. b 8. 1.

Schol, q confirm

We affumed in the demonftration of the laft
Propofition, CDr^BC, becaufe CDq was equal
to BCq : our affumption we prove oy the fol-
lowing theorem*

1 1,

C J THEQ,

;8

The j!rj! Book of,
T HEOREME.

| J

The f quarts AFy CG of $qual right lints ABy CD
art eaual one to the other : And the fides IK, LMof
equal fquares NKy PM> are equal one to the other.

1. Hyp. Draw the diameters EB, HD. Then
a ?4, 1. *s evident triat AF is a equal to the triangle
b + l/&^^yP*et ta^en> anc* * *3ual t0 triangle
6 «. HCD twice t*ken, and equal to a QG. Which

war to he done,

2. Hn. If it may be, let LM be greater than
ai/ , IK. Make LT = IK, and let LS be a equal

?J??- W » W*=NKe=l-Qr
C hyp. ■ w

\ After the fame manner any re<3 angles equila-
teral one to another, are deoionftratod alio to
be equal.

The, End of tie firft Book.

f

I •

i * .

THE

> *

ioogle

THE SECOND BOOK

• T -

OF

*

euclide'/ elements:

- -

■

Definitions.

* *

t

EVery right-angled Parallelogram AB
CD is faid to be contained under
two right lines AB, AD compre-
hending a right angle.
Therefore when you meet with fych
as thefe^thereBande under BA9 AD, or for Jbartnefi
fake the re8 angle BAD , or JiJxJD (or ZA, for
Z x A) the tea angle meant is that which is contained
under the right lines BA> AD fet at right angles.

■

WWfKsiwo complements is csiuea a uiiomou. ■»*
. the Pgr. FB ■+ BIv- GA (EHM) is a Gnomon ; ani
mmf* tU (GKJ) is * Gnomon.

C4 PROP.

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4* Tie fecond Bool of

<S If two tight lines AF, ABy U
~}given, and> one of them AB divi-
ded into as many farts or fegments
as you tleafe ; the reftangle com-

WE"1 ptfo**9* under the two whole
tight lines ABy AF, Jball he equal
to aU the retlangles contained under the whole line
4F and the feveral fegments, ADy BE, EB.

« Set AKtferpendiculair to AB. Thro' F a draw
an infinite lipe FG perpendicular to AF. From
the points, Df E, B eredt perpendiculars DH,EI,
BG, Then is. AG are&angle comprehended un-
b x$.ax.i. dcr AF,AB, and is h equal to the reftangles AH,
« 34» 1. EI, EG, that is (becauie DH, EI,AF c are equal)
to the reftangles under AF, AD, under AF,DE,
under AF, Ed. Winch wot to he dem. V

Schol % *

If two right lines given he hoth divided into lx>w
many farts foever, tie produ3 of the whole multi-
flied into itf elf Jball he tlx fame with that of the
farts multiflied into themfelves. A *

For let Z be =rA-+B^ C, an<i Y =D^E;
then, becaufe DZ a — DA+DB-+DC, and EZ
a t.z. a = EA -+ EB -h EC, and YZ a=TDZ-EZ,
b z. ax. i AallZY be =s DA-+DB +DC+E A-+-EB-+ ECf
Which woe to he dem. ^

From hence is underfioodthe manner of multiplying
compounded right lines into compounded* For you
viujl take all the ReSangles of the farts, and they
will frefent you with the ReEtangle of tjsp wholes.

But whenfoever in the multiplication of lines
into themfelves you meet with tliefe figus — in-
termingled v/i^thefe V, yoij muft alfo have par-
ticular regard to the figns. JFor of multiplied
into — arifeth — ; but of into — atifeth-+,
*x. p. let+A be to be multiplied into B— C \ then
becaufe +-A is not ajBrip^ of all B, but only vt
a part of it, whereby it e^edsC, tfiereforeAC
inuft remain denied ± fo that the produ* will

be

Digitized by Google

»

EUCLIDE'i Elements. 41

be AB - AC. Or thus ; became B confifts of the

parts C and B-C, * thence AB:=AC+A*B-C.* 1. *

take away AC from either,then AB— AC— AxB

C. In like manner if— A be to be multiplied
into B— C, then feeing by reafon of the figure —
that A is not denied ot all B. but only ot fo much
as it exceeds C, therefore AC muft remain affir-
med, whence the produft will be — AB ■-»■ AC
Or thus \ becaufe ABrr AC+AxB — C; take
away all thruughout, and there will be — AB
rr AC —Ax — B * C 5 add AC to either, and
there will be — AB * AC = A x B — C.

This being fufficiently underftood, the nine
following Propofitions, and innumerable others
of that kind, arifing from the comparing of
lines multiplied into thmifelves (wnich you
may find done to your hand in Vieta and other
Analytical Writers) are demonftrated with great
facility, by reducing the matter for the molt
part to almoft a (imple work.

Furthermore, *it appears that the product of* ro,tt#,
any magnitude multiplied into the parts of any
number, is equal to the produft of the fame mul-
tiplied into the whole number : As $ A «* 7 A
se 12 Af and 4 Axj A^4AxyA = 4Ax
iz A. Wherefore what is here delivered of ther-
multiplying of right lines into themfefves, the
fame may be underftood of the multiplying of
numbers into themfelves, fo that whatfoever is
affirmed concerning lines in the nine following
Theorems, holds good alfo in numbers \ feeing
they all immediately depend and are deriv'3
from this firft.

PROP. II.

If a right line Zle di-

^s^00^1 Z vi dei any-wife into two

parts, the reSangles conu
^ - prehended under the whole

ime Zand each of the fegments J, £, arc fyual to
thefquarc viadc of the whole line Z.

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4*

Z I. ft.

The /icon J Book

I fay that ZA-*ZE=:Zq. For take B=Z;
a then is BA-BE-rBZ, that is (becaufe B~ Z)
ZA+ZE~Zq. Wici <» . ^

a 3. 2.

b 2. 1.
c 1.

PROP. III..

If a right line Z he divided
my wife into two parts, the
reclangle comprehended under
the whole line Z and one of the fegments E is equal
to the rt&angle made of the fegments J, E, ana the
fquare defer ihed on the faidfegment E.

I fay ZE^AE + Eq. a For EZ-EA+Eq.

PROP. IV.

// Tight line Z he cut

mZ± ■ z | TS any-wife into two parts, the

t TBf fauare made of the whole line

Z is equal loth to the fquares made of the fegments

J)E,and to twice a reel angle made of the parts A, E.

I fay that Zq— Aq ^Eq + z AE. ForZAtf^t

Aq1-AE,andZEtf- Eq- E A. Therefore whereas

ZA \ ZE b = Zq, thence is Zq^= Aq •+ Eq +

z AE. Which was to he dem.

1 B Otherwife thus ; Upon the
K 1 r: rjg^t j|-ne Ag make the fqUare

AD,and draw the diameter EB ^
through C, the point wherein

tthe line AB is divided, draw
the perpendicular C F ; and
q £ thro' the point G draw HI par
rallel to AB.
Becaufe the angle EHG =A is a right angle,
frtV.Jl* and AEB is half a ri ht> e therefore is the re-

E i 4- ]• fame manner is CI proved to be CBq.
L i9.^/- AG, GD are reftangles under AC, OB. where-
k i^axA. fore the whole fquare AD fe^ACq-h CBq •+ z

ACB. Winch was to he dfm* L „

Cow*

Digitized by

EUCUDE', Elements: ( 4$

CoroU.

I. Hence it appears that the Parallelograms
which are about the diameter of a fquare are
alfofquares themfelves. ^

x. That the diameter of any fquare bilecls its

lCThat if A r= \ Z, then Zq = 4 Aq, and >
Aq -= ^ Zq. As on the contrary, ii ii be to
that Zq = 4 Aq, then is A = ± Z»

PROP. V.

— »-

^ ^ farts AC> CB, and
into unequal parts, AD, DB, the re&angle compre~
hendednnder the unequal fatts AD, BD, together
with the fquare that is made of the intermediate
fart CD, u equal to the fquare thai is made of the
half line CB.
I fay that CBq = ADB + CDq. *

ForthefeareSa C^q+CDB-i-DBq+CDB. • * 4\ *
III equal. jCDq-» h CBD (c ACxBD) J-CDB. £ J* 1#
CCDq <i ADB. j /%

This Theorem is fome^hat differently e%- Q *• *•
prefs'd, and more ealily demonftrated thus ; A
ReBangle made of thefum and the difference of two
lines AyEy is equal to the difference of their f quarts.

For if A + E be multiplied into A-E, there
arifeth Aq— AE \ EA — Eq^ Aq — Eq. % Jfmb
was to he ism.

1 ■ 1 »■] , 1 If the line AB be dir
P £ B B vided otherwife, (aw.?
nearer to the ^©int of bifeftion, in E > Then 1*

A ForTf AEB i=CBq -CEo; and ADB^CBq a j, t. ^
—CDq. Therefore, whereasCDq cr CEq, thence 3. M.
fsAEBcrADB, Wtebvutole**. %

Digitized by Google

44 tt* feconJ Book of

CoroU.

t . Hence is ADq + DBq cr AEq -+ EBq. For
> 4, ADq -h DBq z ADB * = ABq b =: AEq ^«
EBq t AEB. Therefore becaufe x AEB c* z
ADB. thence is ADq -\- DBq tr AEq EB<j,
Which was to be dent.
c 3, «* z. Hence is ADq-V- DBq— AEq +• • + EBq
= z AEB - 2 ADB.

PROP. VI.

If * rlg1>t line A be du
]g ttidi& into two equal parts f
" and another riglrt lint B
Bided to the fame direSly in one right fine, then the
teBdngk comprehended under the whole and the line
added \ (v\z.A + E) and the line added E9 together
with the fauare which is made of 4. the line A% is
equal to the f quart of ~ A+E taken as one lint.
* 4. & ?• 1 % that ^ Aq (a Q: * A) •+ AE Eq = Q?
Cbr. 4.*. TA-»-jE.4For>Q.iA"+E = |Aq-i.Eq-HAE»

GmS.

H#flce it follows that if 5 right lines E* E-+ J A t
E-+A be in Arithmetical proportion, then the
Ke&angle contained tinder the extreme terms E,
£ -+ A, together with the fquare of the difference
•f A, is equal to the fquare of the middle term

PROP. VH.

Jjf df right line1 Z be divi-
ded any-wifc into two parte,
the fquare of the whole line
Z together with the fquare made of one of the feg-
mentt Bf is equal to a double re8 angle comp* tended
binder the whole line Z and the faid fegment E, to-
gether with t1)t fquare made of the other fegment A*
ift I fay fthat Zg Eq = z ZE ^ Aq. For Zq

A^-h £q a AE. and z ZE *s= 2 E^
* AE. r/;i^ to* 40- ** .11.

CoroU.

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EUCLIDE'i Elements. 4$

CoroU.

Hence it follows that the fquare of the diffi*
rence of any two lines Z, E, is equal to the
fquares of both the lines lefs by a dotible reftan-
gle comprehended under the faid lines.

c For Zq -f *<l - * ZEsAq^Q: Z-E. c ?# r ani

P RO P. VIIL l%ax?
If aright line Zbe divided
^any-wift into two farts, tht . *■ H.
A f rtftatiglt comprehended under
th whole lint Z and ont of tht /cements E four
times, togtther with tht J quart of tht othtr ftgmtnt
A, is equal to tht f quart of tlx wholt line Z and
tht ftgmtnt E taken as ont Vine Z+E.
I lay that 4 ZE -\- Aq = Q.: Z + E. Foj

1 ZE a = Zq <+ Eq - Aq- Therefore 4 ZE a
Aq = Zq-Eq ■+ z ZE Z-\- E* JFiici £ **•
wax ro am. j » 4- *«

PROP- IX.

J ■ 1 , > If d right lint AB

A. C D ~Ehe divided into equal

farts AC,CB, and into unequal parts, AD, DB,
then are the fquare s of the unequal parts JD, DB,
togtther, double to tht f quart of the half line AC>
and to tht fquare of tht difference CD.

I fay that ADq+DBq= 2 ACq-*- * CDq. For
ADq-+DBq a = ACq CDq 2 ACD+-DBq.a 4. t;
But 2 ACL) (h 2 BCD) -h DBq crrCBq (ACq) b
CDq. <* Therefore ADq •+ DBq sa 2 ACq ^ c 7-

2 CDq. Winch was to It dem. d 2. a*
This may be otherwife delivered and xriore ea~

fily demonftr ate d thus 1 The aggregate of thtj quarts

made of the fum and the difference of two right lines s

A, E, is equal to the double of tht J quarts m#dt

from thoft lints. , v

For Q; A-+E *rr Aq+Eq-4- 2 AE. and A a
— E * = Aq-t-Eq - 2 AE. Thefe added toge- b - ^
ther roakt 2 Aq ■+ 2 Eq. PTW** f« dm. *

PROP*

Digitized by Google

26 The feemi Book* of

PROP. I.

- ■ • If a tight line A he if-
E mvided into two equal fart s f
and another line he added

in a right line with the fame, then is the fquare of
the whole line, together with the added line, (as be-
ing one line?) together with the fquare of the added
line E, double to the fquare of A and the added-
line E, taken as one line. ^ ,

^ 4. 2. I fay thai Eq ±Qj A + 7. e. a Aq-n 2 Eq + a
bcor. 4.Z.AE = zQ: 1 A -4-2 Q^: | A-E. For z Q: -J.
* 4. 2. A h = i- Aq. Ami z Q: ^ A -4- E c = • Aq hk
2 Eq •+ 2 At. Winch was to he denu

PROP. XL

To cut a right line
°iven A £ in point Gf
) that the reft angle
mprehended under th*

a 45. X.
b 10. 1.

1 1

fegments AG.
the fquare AC. £ BiCeft

c 6. 2.
d cow/rr.
€47. 1.
1 3.

: 1

Upon AB a defcribe
the fide AD in E, and draw the line EB ; from the
line EA produced take EF -EB. On AF make
the fquare AH. Then isAHrrABxBG.

F01 HG being drawn out to 1$ the rectan-
gle DH ^ EAqc-r:EFq^r=EBqe-:~BAq-*-EAq.
Therefore is DH / ~ BAq = to the fquare AC.
Take away AI common to both, then remains
the fquare AH = GC, that is, AGqrrABxBG
Which was to he done.

This Propofition cannot be performed by,
numbers \ ¥ for there is no number that can be
fo divided, that the product of the whole into
one part fhall be equal to the fquare of the other
£>arc»

7 :.v PROP,

Digitized by G

4*

For thefe are
all equal.

EUGLIDE'i Bkmemtl

HOP. XIL

1A In obtufe-angltd triangles, ABC%
T the fquare that is made of the fide
AC/ubtending the obtufe angle ABC
J | is greater than tfje fquares of the
- fides BC, AB that contain the obtufe

cwlc ABC, by a double reftangie contained, under
true of the fides BC, which are about the obtufe ang h
ABC, an which fide produced the perpendicular AD
falls, and under the line BD, taken without the
triangle frovt the point on which the perpendicular
AD falls to the obtufe angle ABC.
I fay that ACq-CBq+ABq -^iCBk BD.
' ACq.

\a CDq U> ADqw a 47. I*

,2> CBq-* 2 CBD -|- BDq -t* ADq. b 4. 1. '
,CBq -* 2 CBD -h c ABq. c 47. *•

Scholium*

Hence, the fides of any obtufe angled triangle ABC
being known, the fegment BD intercepted betwixt the
perpendicular AD and the obtufe angle ABQ, as alfo
the perpendicular it f elf AD JbaU be eafily found out.

Thus- Let AC be 10, AB 7,CB $. Then is ACq
i*o,ABq 49,CBq 2$. And ABq+CBq=74. Take
that out of ioo, then will 26 remain for 2 CBD.
Wherefore CBD lhall be 125 divide this by CB
5, there will 2 f be found for BJD. Whence AD .
will be found out by the 47. 1.

PROP. XIII. 1 • » '

In acute- angled triangles ABC,
the fquare made of the fide AB
fubtending the acute angle ACB,
is lefsthan the fquares made of
the fides AC, CB comprehending

Qthe acute angle ACB by a double

reel angle eoniained under one of
the fides BC, which are about the acme angle ACB
on which the perpendicular AD falls, and under tb$. .
line DC taken within the triangle from the perpendi-
cular 4D t9 tit mfr j

Aft J

I

Google

4*

a 47. t.
b j.'i.
c 47. It

The fecmi $iol eft &C.

I fay that ACq -4- BCq ■= ABq ■+ z BCD.
1 ACq -+ BCq*
For thefe are) a ADq + DCq + BCq.
equal. ) b A Do BDq -4. z BCD*

OABqi 2BCD,
CbtoB.

Hence, TA* jWe* o//in acute-angled triangle ABC
being knoren,yu may find out thefegment DC inter*
cepted betwixt the perpendicular AD and the acute
angle ACB, ai alfo the perpendicular it felf AD.

Let AB be 15, AC is, BC 14. Take AB<i(i6o)
from ACq -+ BCq, that is, from 22 5 +196—411.
Then remains 252 for 1 BCD. wherefore BCD
1vill be 126. divide this by BC 14, then will 9
be found out for DC. From whence it follows
AD = V- "J - 81 as 12.

PROP. XIV.

i*|5 >

b ro. U

♦46.1

c conftr.

to find a fquare ML equal to a right-lined figure
given A.

a Make thereftangle DB=A, and produce the

freater fide thereof DC to F, fo that CF = CB.
bifed DF in G, about which as the center at
the diftance of GF defcrihe the circle FHD,and
draw out CB till it touch the circumference in
H. Then {hall be CHq = * ML =3 A. \
For let GH be drawn. Then is A c = DB c
dj. z.andr=zDCXd = GBq - GCq « = HCq c = MU-
} . ax. Which was to be done.
* 47. 1.4*4

Tht End of tU frond Book,

I

THE

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THE THIRD BOOK

O F

EUCLlDE'j ELEMENTS.

• » < r

Definitions.

Qual circles (GAKC, Ht>EF) are
fuch whofe diameters are equal ;
pr, from whofe centers right lines
drawn GA, HD, are equal.

II. A right line AB is faid
to touch a circle 1?ED, when
touching the fame, jmd being
produced, it cuttcth it not.

The right line FG cuts the
circle FIT).

« ■

III. Circles DAC, ABE (and alfo FBG,ABE)

D i ai«

The third Book of

are faid to touch one to the other, which touch,
but cut not one the other.
The circle BFQ cuts the circle FCH

IV. In a circle GABD,
tight lines FE,KL are faid

. to be equally diftant from
\ the center , when per-
ik^pendiculars G H , G N
J drawn from the center G
to them are equal. And
that line BC is laid to be
furtheft diftant from it, on
whom the greater perpen-
dicular GI falls.

V. A fegment of a cir-
p cle (ABC) is a figure con-
tained under a right line
AC, and a portion of the
circumference of a circle
ABC.

VI. An angle of a fegment CAB, is that an-
gle which is contained under a right line CA
and an arch of a circle AB.

VII. An angle ABC isfaid to be in a fegment
ABC, when in the circumference thereof fome
point B is taken, and from it right lines AB,
CB, drawn to the ends of the right line AC,
which is the bafe of the fegmetit ; then the an-
jle ABC contained under the adjoined ljnes AB,
;B, is faid to be angle in a fegment.

VIII. But when the right lines AB, BC com-
prehending the augle'ABC, do receive any peri-
phery of the circle ADC, then the angle ABC
is faid to ftaod upoa that periphery. ,

4 «

IX.

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EtfCLtDE'i Elementi.

IX. A feflor of a circle (ADB)
is when an angle ADB is fee at
the center Dot chat circle ; name-
ly, that figure ADB, comprehend-
ed under the right lines AD, BD
containing the angle, and the part
of the circumference received by
them AB,

X. Like fegments of acircle (ABC, DEF) ar«
thofe which include equal angles (ABC, DEF;)
or, in whom the angles ABC, DEF are equal.

PROP, h

to find the center F of *
circle given ABC.

Draw a right line A C
any- wife in the circle, which
bifeft in E, thro1 E draw a
perpendicular DBj and bi-
r feci the fame in F, the point
'Cf fhall be the center.

If you deny it,let G a point
without the line DB be the
center, (for it cannot be in the line BD, fince
that cannot be divided equally in any point but
FO let the lines GA, GC GE drawn; Now if
G be the center, a then is GA-rGC, and AE— a ijJcfA.
£C by conftruition, and the fide GE common, bb 8. I.
Therefore are the angles Cj,EA,GEC equal, andc loJef.u
c confequently right, d Therefore the angled 12. ax.
UEC:xFEC. e Which is abfurd. e 9. «.

D 2 CoroU.

niniti7PH h

y Google

j 2 The third Book of

Coroll,

Hehce, if a right lineBD bifeft any right
line AC in a circle at right angles,; the center
fhall be in the line BD that cuts the other*

i

Jndt* Ti^ center of a circle is eafily found out hy applying
Tacq. th top of a fquarej^jo the circumference wereof.For
if the right line DE that joins the points D, E,
in which the fides of thefqtiare QD, QE cut
the circumference, be bifefted in A, the point A
fhall be the center. The demon ftrat ion whereof
depeids upon Prop. ji. of this Book.

PROP. II.

// in the circumference of a cir-
cle CAB any two points J, B It
taken, the right line AB which
^yoinsthofe two points fiall fall
I within the circle.
J Take in the right line AB
any point t> ; from the center
a r fufe/.t.C draw CA, CD, CB- Becaufe CA a — CB,
b 5. i. therefore is the angle A h B. But the angje
c 16. r. CDB f r A, therefore is CDB c B, therefore
d 1 p. r. CB d c CD. But CB only reaches the circumfe-
rence, therefore CD comes- not fo far ; where-
fore the point D is within the circle. The fame
itiay be proved of any other point in the line AB.
"And therefore the whole line AB falls within
tte circle, Which was to ieAenu

*€oretl:

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EUCLIDE'i Elementi. ^

Coroll.

Hence, if a right line touch, a circle, To that
k cut it not, it touches but in one point,

p itop. m.

* ■

// in a circle EABC a right '
line BD drawn thro* the cenur,
h[e& any other line AC not
drawn through the center ^ it [0 all
alfo cut it qf right angles in F:
And if it cuts .it at fight
angles y it ft all alfo bi/ed the
fame.

From the center E let the
lines £ A, £ C be drawn.

1. Hyp. Becaufe A F a = FC, and EA ^.r: a ju*
EC, and the fide EF common ; the angles EFA, D //frfa
EFC c fhall be equal, and d confequently right, c g/j. '
Which was to be dem. <j t9jefu

Hyp. 2. Becaufe EFA e = EFC, and the angle e ;w
EAF / -35 ECF, and the fide EF common 5 g ix{ax.
therefore is AF—FC. Therefore is AC cut intof 1 I#*
two equal parts. Which was to be dent. g ^

Coroli. 8
Hence, in any equilateral or Ifofceles trian-
le, if a line drawn from the vertical angle hi-
ed the bafe, that line is perpendicular to ir.
And on the contrary, a perpendicular drawn
from the vertical angle bifects the bafe. . *

PROP. IV.

6

If in a circle ACD two right
lines A By CD cut thro* one
another^ yet neither of tlxem
pafs thro* the center E9 then
neither of thofe lines are divi-
ded into equal parts.

For if one line pafs thro*

Digitized by Google

w

J4 Tht third Book 9f

the center, it appears that it cannot be bifefled
by the other ; becaufe by Hypothecs, the other
does not pafs thro9 the center.

If neither of them pafs thro' the center, then
from the center E draw EF : now if AB, CD
- « , were both bifeded io F, then a would the an-

r • Jy equal, b Which u abfurd.

PROP. V.

* If two circk* BJC,
B DC cut one the other,
they jball not have the
fame center E.

For otherwife the lines
£ B, LDA drawn from
E the common center ,
_ would DE be a — EB

ViJef.t. ^~ tT a = EA. * Which is

PROP. VI.

If two, circles B 4 C, B D E9
inwardly touch one the other (in
B) they have not one and the
fame center F. J

For otherwife the right lines
FB, FDA drawn from the cen-
ter F» would be FD a r= FB
a FA. h Which is abfurd.

./•■

9 *A ;

.«

PROP..

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EUCLIDEV Elements.

*

SS

PROP. VII.

If in A B the diameter

/\ I *f a cm^ fome pot** 0

\ I \ take 7i9 which is not the center
\\l A of the circle, and from that
^ ' . point certain right lines GC,
CD, GE fall on the cirr
tie, the greateft tine Jball be
that (GJ) in which is the
center F * the leaft, the re-
mainder of the fame line (GB)
And of all the other lines* the line G C neareft to
that &'bich was drawn thro* %he center is always
greater than any line farther removed GD ; and
only two lines are equal GE, GH, wh.ch fall upon
the circle from the fame point, oh each fide of tk
leaft GB or of the greateft GJ.

From the center F draw the right lines FC,
FDf FE ; and a make the angle BFH—BFE. a
t. GE -1 FC (that is GA) a rGC. Which a
was to he 4cm.

z. The fide FG is common, and FC* t= FD,b
and the angle GFC c cr GFD ; d wherefore the c
bafe GC cr HD. d
FB <FE) e -3 GE -j- GF. /Therefore FG,e
which is common, being taken away from both.f*
BG-3EG.

4. The fide FG is common, and FE~FH,and
the.angle BFH^ rrz BFE ; b Therefore is GE— g
jGH. But that U6 other line GD from the point h
Q can be equal to GE, or GHf is already pro-
\£d. Which was to be dem.

I.

10. I.

9. ax. r
24. 1.
zo.
5. ax.

conjlr

4ft.

/ * ...

U.x. ■

D4

PROP.

t * -

Digitized by Google

The third Book of
PROP; VIII.

;'»vi.

If fome point A he
taken without a circle,
and from that point he*
drawn certain right lines
M, AH, AG, AF to the
circle, and of thofe one
Al he drawn through the
center K, and the others
any-wife ; of all thofe
lines that fall on the
concave of the circum*
ference , that is Hie
greateft A I which is
drawn through the cen>
ter ; and of tlx others,
that which is near eft {AH) to the line that paffes
through the center is greater than that which is more
diftant AG. But of all thofe lines that fall on tBe
convex part vf the circle, the leafi is that AB
which is drawn from the point A to the diameter
IB ; and of the others, that ( AC) which is nearejt
iotheleaft, u f iefs than that which is farther diftant
AD. And pom that point there can he only two
equal right lines AC, AL drawn, whicFJhall fall on
the circumference on each fide of the leafi line AB or
of the greateft AL

From the center K draw the right lines KH;
K<3> KF, KG, KD, &$. and make the angle
AKLr^AKC.
. . i. AI ( AK r|- KH ) a zr AH.

z. The fide AK is common, and KH = KG,
h M' *• and the angle AKHrAKGj h therefore
the bafe AHir AG.

5. KA £ -1 KC -|- CA. From hence take
away KC, KB that are equal : then will remain
A3 d -3 AC.

4. AC CK e -3 AD -\- DK. From thence
take away CK, DK that are equal 5 then re*
mains AC/ -3 AD,

$. Tha

a ao. t*

C ZO.I.

d $. aoe*
t zi. u

1

Digitized by Google

EUCLIDE'j Elements. S7

5. The fide KA is common, and KL z=z KC,
and the angle AKL g^ AKC \ h therefoieg con/tr.
LA srrtxCA. But that no other line could n 4. X*
be drawn equal to thefe, was proved above.
Therefore, &c*

PROP. IX.

Jf in a circle BCK a point A
he taken, and from that point
more than two equal right lines
AB, AC, JK9 drawn to the cir-
cumference, then if that point A
the center of the circle.

For a from no point with- a 7* 3«
out the center can more than
two right lines equal be drawn
the circumference. Therefore A is the ceji-
■. Which was to he dem.

to

ter,

PROP. X.

A circle I A KB L
cannot cut another cir-
cle IEKFL in more
than two points.

Let one circle, if it
may be, cut the other
in three poi»ts,I, K, L,
and I K, KL being
iojn*d, let them be bi-
fefted in M and N.
a Both circles have a cor* *• I*
their centers in their perpendiculars MC, NH,
and in the interfe&ion of thofe perpendiculars, ,
which is O.JTherefore the circles that cut each b *• 3*
other have the fame center. Which is falfe, '
Prop. S.h '

PROP,

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PROP. XL

r

is-

i
t

If two circles GADM,
F AB C touch one the
other inwardly \ and their
centers be taken (?, Fj
a right line FG joining
their centers^ and po-
duced, Jball cut tU cir-
cumference in A tlx point
efcontaB of the circles.

If it can be, let the
right line FG produ-
ced cut th« circles in fome otjier point than Aw
fo that notFGA,but FGDB (hall be a right line,
a x$.A/.if Ifttthe lmeGA bedrawiu Now, becauTe GD<i
h 7. 5. ==:GA, and GB*~GA (finqe the right line FGB
paffes through F the center of the greater circle)
$9. ax, 'tercfore U GB x> GD. c Wliich is tbfurd.

PROP. XIL

a 20. u

// two circles ACD, %QE touch one the other cut-
ffdly tlje right linp AB which joins their center 's
J, B, JhaUpafs throy the point of contaS C.
m If it may be, let A DEB be a right line cut.
ting the circles not in the point or conraft C,
but in the points D, E draw AC, CB. then is

PROP.

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EUCLIDE'* Elements^

59

PROP. XHf f

4 Circle
C A f cannot
touch a circle
B A H in more
points than one
Ay whether it
he inwardly or
outwardly. *

!« Let one
Circle (if it can
be) touch ano-
ther in two

?oints A, H, a a !!• ?i
'hen will the
right line C B
$ that joins the

centers, if produced, fall as well in A, as H.
Now becaufe CH *~CA, and BHcCH, there- b r$Jef.u
fore is BA (c BH) c CA .d Which is aifitrd. c i sJef.x.

z. If it be fa id to Jouch outwardly in the points d o. ax*
E and F, then draw the line EF, • which will e 2. %%
be in both circles, Therefore thofe circles cut
• ' Other 5 Which is again* the Hyp.

PROP. XIV.

In a circle EJBC equal
right lines ACX B D are c-
qually dijlant from the center
p: and right lines AC, BD
which are equally d fiant from,
the center, are equal among
themj elves.
,r Fipm the center E draw

C a the perpendiculars EF, EG.
a which will bifea the lines AC,B£>. join EA,EB. a 3. ^
iJfyf. AC^BD. therefore AF^££. Butal- b 7- ax.

Digitized by Google

<5d The third Bool of

fo EA—EB. therefore EFq EAq— AFq ~
c EBq— BGq c zz EGq. d Therefore FE = EG.

3. ax. i.Hyp. EF- EG.Therefore AFq<r=EAq— EFq
d/^.48.i.=EBq— EGq^BGq. Therefore AF <fc=2GB,and
c 6, ax. e confequently AC zLBD. Which was to be dan.

PROP. XV.

In a chde G A B C
the greateft line is A D
the diameter \ and of all
otlm lines, that tine PE,
which is nearefi to the
center G is greater than
any line BC farther diftant
front it*

i. Draw GB and GC.
The diameter A D ( a
GB+GC) b l-BC.
h. Let the diftance GI be ir GH. TakeGN=;
GH. Thro* the point N draw KL perpendicularly
to GI: JoinGK, GL. BecaufeGK~GB,andGt
=r GC, and the angle KGL r-BGC ; c there-
fote is KL (FE) tr BC. Wbieb was to be dem.

a i $Mf.u * 1
b 20. i.

r 1

C 24, 1.

• *

PROP.

■- -

A bine C D
-drawii from the
extreme point of
the diameter H A
of a circle BALH,
perpendicular to the,
[aid diameter y Jball
fall without the
circle \ and between
the fame right line
and the circumfe-
rence cannot be
drawn another line
AL. 4nd the avgle

Digitized by Google

EUCLIDE'* Eletnentf. 61

of tie femicircle BAIy is greater than any rigU-ftnei
ticute angle BAL \ anH the remaining angle<without
the circumference DAI is Lefs than any right-lined
angle.

1. From the center B to any point F in the
right line AC, drawthe right line BF. The fide

BF lubtending the right angle BAF is a greater a t£< t.
than the fide BA which is oppofite to the acute
angle BFA. Therefore whereas BA (BG ) reaches
to the circumference, BFfhall reach further; and
fo the point F. and for the fame reafon is any o-
therpointofthelineACplacedwithout the circle.

2. Draw BE perpendicular to AL. The tide

BA oppofite to the right angle BEA is h greater b ip« U
than the fide BE which fubtends the acute angle
BAE ; therefore the point Ef and fo the whole
line EA falls within the circle.

Hence it follows that any acute angle, to
wit, EAD, is greater than the angle of contail
DAI.and that any acute angle BAL is lefs than the
angle of a femicircle BAL Wlmhwaitobedenu

Coroll.

Hence, A right line drawn from the extre-
mity of the diameter of a circle, and at right
angles, is a tangent to the faid circle.

From this Propofition are gathered many P*.
radqx and wonderful Gonfedtaries, which you
may meet with in the Interpreters.

PROP. XVIL

From a point given A
draw a right line r AC which
JbaU touch a circle given DBC*
From D the center of the
circle given bt a line DA
cutting the circumference in
B, be drawn to the point
giTen A ; from: the center D
defcribe another circle thro*
A ; and from B draw a perpendicular '
? which ftalL meet with the circle AE in '

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& • the third Book of

the point E \ and draw ED meeting with the
circle BC in the point C. Then the line drawn
from A to C fhall touch the circle DBC.
a uJef.i. For DB a — DC, and DE a r= DA, and the
b 4. I. angle D is common; b therefore the angle ACD
c w.16. 3. EBD and right, c Therefore AC touches the cir-
cle in C. Wlich was to fa done.

PROP. XVIIL

• If any right line AB touch
a circle FEuC> and from the
center to the pomt of conta&
E a right line FE be drawn$
that line EE fhall be perpendi-
cular to the tangent AB.

If you deny it, let fome
other line F G be draton
from the center F perpendi-
cular to the tangent, and
a z. def. j. a cutting the circle in D; Therefore, whereas
t6$. the angle. FGE is faid to be right, b thence is
bcor.i7.i.tJie ande FEG acute ; c fo that FE (FD) cr
c 19. 1. FQ. Winch is abfurd.
d 9. ax. t

PROP. XIX.

// any right line AB touch
'a circle, and from the point of
contaft C a right line CE be
oe3ed at right angles to the
tangent, the center of the
circle Jh all be in the line CE
fo eretfed.

If you deny it; let the
center be without the line
CE in the point F ; and from F to the point of
• 18. J; contact let FC be drawn. Therefore the angle
a 11. ax. *FCB is right,and a confequently equal to the
b 9; ax. angle ECB, which was right by Hyp. I Which
is abfurd%

frROfc

EUCLIDE'f Elements.

PROP. XX.

In a circle DABC, the angle BDC at the center is
doulle of 'the angle B AC at the circumference, when
the fame arch of the circle BCu the bufc of the angles.

Draw the diameter ADE. The outward angle <«ffc
BDE a - DAB -f DBA h — z DAB. LikewiU- a 5. t.
the angle EDC — z DAC. Therefore in thefirft b 5. 1.
cafe the whole angle c BDC = z BAC. and in c z. ax*
the third cafe the d remaining ajigte BDC — t d 20. <wr.
BAC. Which was to be dcm. * T

PROP. XXI. V

In a circle ED AC, the angles DAC and DBC
wheh are m the fame figment, me equal one to the
other.

1. Cafe. If the legment DABC be greater than a
femicirde.from the center E draw ED, EC. Then
is twice the angle A a—Ed— zB. W.W.tobedcm. a *o. ?«

iXafe. If tlic legment be lei's thanafemicircle,
then ijthe fumot the angles of the triangle ADF
equal to thelum of the anglesof.the triangle hCt.
fcom $ach let AFD be taken away b equal to b t<. r.

BFC,

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c t. ax.

64 The third Book of

c By the U BFC, and ADB <r—ACB be like wife taken away,
eafc* then remains DAC^DBC. W. W% to he dem.

PROP. XXTI.

The angles ADC,
ABC of a quadrilateral
figure ABCD defcribed
in a circle, which are
oppofite one to the other,
are equal to two right
angles.

Draw AC, BD.
The angle ABC f
BAC a — z
right- But B DA b~
BCA, and BDC b = BAC. c Therefore .BC -4-
ADC = 2 right angles. VHnchwas to be dem.

CoroU.

* See the *• Hence, If one fide* AB of a quadrilateral
following defcribed in a circle be produced, the external
Diqgr. anSle EBC is equal to the internal angle ADC,
which is oppofite to that ABC which isadjacent
toEBC, as appears by 13. 1. and ax.

2. A circle cannot be defcribed about a Rhom-
bus ; becaufe its oppofite angles are greater or
lefs than two right angles.

Schol.

If i7i a quadrilate-
ral ABCD the angle*
A a7\d C, which are
oppofite , be equal to
two right, then a circle
may be defcribed about
that quadrilateral.

For a circle will
pafs thro* any three
angles (as fhall ap-
pear by 5. 4. ) I fay
that fliall pafs thro' A the fourth alfo of fuch

a

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EUCLIDE'i Elements. 6$

a quadrilateral: For if you deny it, let the cir-
cle pafs thro' F. Therefore the right lines BFf
FD, BD being drawn, the angle C ■+ F a = i a n. j.
tight b — C -+ A wherefore, A c is equal to F. b

d Jr&VA fx abfurd* c J. m«

d at. i.

PROP. XXIII.

Two like and unequal fig-
ments of cinles ABC, ADC
cannot be fet on. the fame
right line AC, and the Jame
"q*1 fide thereof.
For if they are faid to be like, draw the line
CB cutting- -the circumference in I) and B*
join AB and AD. Becaufe the fegments^are
luppofed like; a therefore is the angle ADC= MCwfij
^ ABC. b Which h abfm rA b l& !♦

PROP. XXIV*

Like fegvtentt

/^JT\ /<£>v */ circles A B ft
Y A / \ 25£F ix/>o» *jti<i/

|g J|gy ■ U r#/;f lines AC%

«, D/7, are equal one

to the other.

The bafe AC
bein^y aid on the

CFAD CF bafe DF will a-
gree with it, becaufe ACrrDF. Therefore <he
legment ABC fhall agree with the fegment DEF
(for otherwife it fhallfail either within or with-
out, and if fo a then the fegments are not like,*
which is contrary to the Hypothecs, and at leaft
it (hall fall partly within and partly without,
and fo cut in three points, b which is abfurd. c b to. }.
Therefore the fegments ABC s& DEF. Which ^ 8.**.
*w to be dent.

■

t PROP.

The third Bool of

PROP. XXV.

A fluent of a circle ABC
leing given, to defcrihe tlx
whole circle whereof that u a
fegment*

Let two right lines be
drawn AB, BC, which bi-
feft in the points D and E<
From- D and E draw the perpendiculars DF, EF
meeting in the point F. I fay this point fliall
be the center or the circle.

For the center ihall be as well in a DF as ;
FE, therefore it muft be in the common point
F- Which was to he done,

PROP. XXVL

* •

tn equal circles GABC, HDEF, equal angles
ftand upon equal parts of the circumference, A C,
DF ; whether thofe angles he made at the centers ,
<5, J/, or at the circumferences ,2?, E.

Becaufe the circles are egual, therefore is G A
= HD, andGC = HF; alfo by Hypothefis
t±t. the angle G = Hj a therefore AC = DF.
b za j. Moreover the angle BirriQcsr^H h = E,
c typ. d Therefore the legments ABC, DEF are like,
d io.defi* and e confequently equal, f whence the remain-
e 24. j% ing fegments alfo AC, DF are equal* Winch
f 5. ax* was to he dem.

1 •

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EUCLIDE'/ Elements.

*7

CorolL

In a circle ABCD let an arch
P AB be equal to DC 5 then mall
AD be parallel to BC. For the
right line AC being drawn, the
angle ACB a 5= C AD ; where- a 26. J.
fore by 27. 1. the faid tides are
parallel.

P.ROP. XXVIL

In total circles
GABC, HDEb\ the
angles Jlanding ufon
equal tarts of the
circumference A C,
jp D F, are . equal be-
tween t he mf elves ,
whether they be made at the centers G, H, or at the
circumferences , B, £.

For if it bepoffible, let one of the angles AGC
be cr DHF,and make AGIrrDHF. thence is the
arch AI a = DF b = AC. JPfocfc if fltyW. a 26. J,

b

Schol. c 9. tf*.

X

^ right line EF, which
being drawn from A. the
middle fomt of any peri-

circle, is parallel to the
right line BC fub tending
q I the faid periphery.

From 1 he center D
draw a right line DA to
the point of contact A,
and join DB, DC.
The fide DG is common, and DB=DC, and
the angle BD A a=r CD A>becaufe the arches BA, a 27. 3.

£ 2 CA

Google

68 the third Book of

b hyp. CA are b equal) therefore the angles at the bafc
c 4. 1. DGB, DGC are c equal, and d confequently
d icufc/li.right j But the inward angles GAE, GAF are
thyp. alio e right, /therefore BC, EF are parallel,
f 28. 1. Which was to be dem.

PROP. XXVIIL

— ■ am

In equal circles
GABC, HDEF,
equal right lines AC,
DF cut off equal farts
po/ the circumference,
rthe great eft ABC e-
gual to the greateft
DEF, and the leaft AIC to the leaft DKF.
From thecentersGtH draw G A GC,&HD,HF
a fyp. Becaufe GArrHD, and GC ~HF, and AC a
b 8. 1. — DF, h therefore is the angle G^H ; c whence
c z6. 5. the arch AIC-=:DKF ; <*andfo the remaining
d j. ax. arch ABC = DEF. Which was to be dem.

But if the fubtended line AC be cr or -3 than
DF, then in like manner will the arch AC be
cr or -p than DF.

PROP. XXIX.

a hyp.
b 2,7.
c 4. r.

In equal circles
GABC, HDEF, equal
right lines AC, 1) F
fubtend equal periphe-
ries ABC? DEF.
Draw the lines
GA, GC, andHD, HF. Becaufe GA^HD,
and GC = HF, and (becaufe the arches AC, DF
are a equal) the angle G b ~ H. c therefore is
the bafe AC = DF. Winch was to be dem.

This and the three precedent Propofitionsmay
be mideiftood alio of the fame circle.

PROF.

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EUCLIDEV Elements. 69

PROP. XXX.

To cut a periphery given JSC
into two equal parts.

Draw the right line AC, and
bifeftit in D \ from D draw a
X> £J perpendicular D B meeting
with the arch In B, it fhall bifeel the fame.

For join AB, and CB. The fide DB is com-
mon, and AD a—DC, and the angle ADB b •= a covjlr.
CDB, c therefore AB ^ BC> d whence the arch b tz. ax.
AB =: BC. Winch was to be done. c 4. i.

d 28. 3.

PROP- XXXI.

In a circle, the angle
JBCy which is in the femi-
cide, is a right avgle ;
but the angle , which is in
the greater fegment BAC
is lefs than a right avgle ;
and the angle which is in
the leffer fegment BFC is
greater than aright angle.
Moreover, the angle of the greater fegment is greater
than a right angle, and the angle of the leffer feg-
ment is lefs than a tight angle.

From the center D draw DB, becaufe DB m
DA, therefore is the angle AarrDBA, and the a $. 1.
angle DCB rtrrDBC, b therefore the angle ADC b 2. ax.
= A ACB c — EBC, d fo that ABC and EBC c 52. 1.
are right angles, f^. W. to be dem. e Therefore d t^Jef.x
BAC is an acute angle. flK W. to be denk And e cor.tj\&\
further, whereas BAC/-»- BFC^r z right,there' f 22. 3.
fore BFG is an obtufe angle. Laftly, the angle
contained under the right line CB, and the arch
BAC is greater than the right angle ABC ; but
the angle made by the right lineCB and the peri-
phery of the lefler fegment BFC g is lefs than g 9. ax
the right angle ABC. Wlmh was to he dem.

B 3 Schol.

7°

• ■

a z6f j.
b 19. 5.
c 5/.
d 52. 1.
c confer*
f 3. ax.

22. $.
k J. d*.

2

Tie third Book of ;

Jn a right-angled triangle A&C* if tie Hypotbe-
vufe (or J lib tended line) AC be Hftffcd in U, a cir-
cle drawn pom. the center D thro' the pojnl-Ajball
alfo {a ft thro' the point B : As you way ekjiiyj^-
monjliatc from this Prop, and 21*1.

PROP. XXXII. -Jr — .>
£ If a right lim A *

touch a circle, andfrqm
vs the point of contaa be

right line CE
.Vr cutting the circle, the an-
' gles ECBy EC A which it
mqkes with the tangent
li?ic are equal to thofe an-
gles EDC> EFC which
are made in the alternate
fegments of the circle.
_T the fide of ^he angle EDC, be per-
pendicular to AB (a for it's to the fame purpofe)
b therefore CD is the diameter, c therefore the
angle CED in a fenvcircle is a right angle, d and
therefore the angle D -+ DCE — to a right an-
gle e - £CB + DCE. /Therefore the angle D
rr= ECB. Which was to be devi.

Now whereas the angle ECB ECA g — 2
right h — D , F, from both of thefe take away
ECB and D, which are equal, k then remains
ECA = F. Winch was to be dem.

PROP.

XXXIII.

Upon a right line
AB to defer ibe a
figment of a circle
AIEB which JbalL
contain an angle
A I B equal to a
fight lined angle
wen C.

a Make the angle
BAD

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EUCLIDE'i Elements.

BAD C. Thro' the point A draw the line AE
perpendicular to HD. At one end of the line
given AB make an angle ABB 2= BAF, one fide
whereof let it cut the line AE in F \ from the
center F thro* the point A, defcribe a circle
which mall pals thro'B (becaufe the angles FBA
I zr. FAB. and c therefore FB - FA, ) AIB conflr.
is the fegment fought. For becaufe HD is per- c o. 1.
pendicular to the diameter AE, it therefore d* cor. ic
touches the circle HD which AB cuts. And*; 31. ?.
therefore the angle AIB c^BAD/=: C. WlnclA f«F*

was to be done.

PROP. XXXIV.

. From a circle given
ABC to cut off a fegment
jBC containing an angle
B equal to a right-lined
angle given 1).

. a Draw a right line a 17. ;
p IF which fhall touch the b 13. 1

circle given in At Met c 32.1
AC be drawn alfo making an angle FAC— D. d conft
This line fhall cut off ABC containing an angle
B c — CAF d = D. Winch was to he done.

PROP. XXXV.

circle FB
C A two
right lines
ABpCcut
each other 9
the retlan-
glecomfre-
bendcaun*
der the feg.
-mertts Ac9
EB of the
&ie ,

he equal to

the .

the third Bool of

the reRangle eompehended under the fegments CE9
ED of the other.

1, Cafe. If the right lines cut one the other
in the center, the thing is evident.

2. Cafe. If one line AB pafs thro* the center
F, and bifeft the other line CD, then drawFD.

* »• *• Now the reclangle AEB+FEq a FBq *=:FDq c
b/<r*.4».i.- EDq-4-FEqd ~ CED-hFEq. e Therefore the
c 47. 1, reftangle AEB -CED. Winch was to be dem.

hp* I. Cafe. If one of the lines AB be the diameter,
e j. ax. and cut the other line CD unequally, bifeft CD
by FG a peipendiculai from the center.
r The rectangle AEB ■* FEq.

f *. *• There V FB(1 (¥D{;Q &

g 47- *• are e- <g FGq - GDq.

h 5- *• cAualt )hGq -+ 7; GEq ■+ Re&ang. CED.

k47-i. O FEq - CED.

1 J. <J*. / Therefore the Reftangle AEB = CED.

a. Cafe. If neither of trie right lines AB, CD
Is thro' the center, then thro* the point of in-

pa!

terfe&ion E, draw the diameter GH. By that
which hath been already demonftrated, it ap-
pears that the reftangle AEB = GEH = CED.
Winch was to be dem.

More eafily, and gene-
rally, thus; join AC and
BD, then becaufe the an.

hlVl' f \ gks a CEA> D£B, and

0 * \ b alfo C, B (upon the fan*

rjJkf1fl Cft^ - / arch AD) arecqual, thence

cw-*w' SCSfc v / are the triangles CEA BED

j * iVV V £ * equiangular. J Wherefore

« £ * ^ CE» ?: EB> ED. and 0

confequently CExEDss
A E x E B. Which wot to be dem.

The citations out of the fixth Book, both
here and in the following Prop, have no de-
pendence upon the fame; ; To that it was free
toufetheiiu *

PROP.

Google

EUCLIDE'/ Elements,
?)\Of. XXXVI,

n

If any point D be taken without a circle E B C,
And from that point two right lines D A, D B
fall upon the eircle, whereof one D A cut the cir-
cle, the other D B touches it, the reftangle com-
prehended under the whole line D A that cists the \
circle, and under D C that part which is taken
from the point given D to the convex of the
periphery, JbaU be equal to the fquare made of the
tangent line.

i. Cafe. If thefecant AD pafs thro' the cen-
ter, then join EB, this a will make a right an- * 18. li
gle with the iine DB, wherefore DBq -+ d EBq b 47- *•
(ECq) b = EDq e r= AD* DC ■+ ECq. There- c 6. z.
fore AD x DC =: DBq. Winch was to be dem. d 3. ax.

z. Cafe. But if AD pafs not thro* the center
then draw EC, EB, ED, and EF perpendicular
to AD, a wherefore AC is bifedted in F. a j. j.

Becaufe BDq -4. EBq b =: DEq b s£ EFq -|- b 47- 1.
FDq c = EFq ADC -* FCq d ^ ADC i c 6. 2.
CEq (EBq.) e Therefore js BDqzrAQC. Which d 47. r.
to be dem* e }. ax.

Merc

The third Book of

More eafilr, and gene-
rally, thus; Draw A§ and
B C. Then becaufe the
angles A, and DBC a are
equal, and the angle D
common to both, thence
are the triangles EDC,
A D B b eqiiiargular.
c Wherefore AD. I)B ::
DB.CD. and dconfequent-
ly AD x DC r- DBq,
Which was to he dan.

CoroII.

1. Hence, If from any point
A taken without a circle, there
be feveral lines AB, AC drawn
which cut the circle, the rectan-
gles comprehended under the
whole lines AB, AC, and the
outward parts At, AF are e-
[qual between themfelves.

For if the tangent AD be
J/ drawn, then is CAF = ADq a
C =BAE.

2. It appears alfo from
hence, that if two lines A B,
A C drawn from the fame
point do touch a circle, thofe
two lines are equal one to the

JO other.

For if AE be drawn cutting
the circle, then is ABq a r=
EAF b - ACq.
J. It is alio evident that from
a point A taken without a circle, there can be
drawn but two lines AB, AC that fliall touch
the circle.

For if a thiid line AD be faid to touch the cir-
cle,thence isADc = ABczzJlCJ Winch is abfurd.

4. And

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7i

EUCLIDE'f Elements.

i

4, And on the contrary, it is plaiq, that if
two-equal rigjit l|nas AB, AC fall from any
point A upon the convex peripheiy of a circle,
and that if one of thefe «jual lines A 6 touch
the circle, then the other AC touches the circle
alfo.

For if pofTible, let not AC, but another line
AD, touch the circle ; therefore is AD c z=. AC e Z*corx
/AB. g Which is atfurd. f byp.

PROP. XXXVIII. g8.Jt

If without a arcle EBP any
point D be taken, and from that
point two right lines DA, DB
fall on the circle, whereof one
line DA cuts the circle the other
\T)B falls uton u ; and if alfo
'ie reft angle comprehended under
>e whole line that cuts the cir-

cle, and under that part of ic
DC which is taken betwixt the

pi

A ointD and the convex periphe-
, , be equal to that [quart which is made of the
me DB falling on the circle, I fay that the line DB
[0 falling JbaU touch the circle given.

From that point D a let a tangent D F be a 17. j%
drawn, and from the center E draw ED,EB,EF.
Now becaufe DBq b = ADC c — DFq, there- b hyp.
fore is DB dzzrDF: But EBrz EF, and the fide c 36, J.
ED common 5 e therefore the angle EBDrzEFD. d jjix.j*tf
but EFD is a right angle, and /therefore EBD fch. 4* U
is right alfo, and g therefore DB touches the e 8. 1,
circle. Which was to be dem. a cor.i6*Z*

\ Coroll. b 8. It

From hence it follows that the h angle EDB

The End of th$ third took*

THE

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THE FOURTH BOOK

o F

EUCLIDEV ELEMENTS.

Definitions.

Right-lined figure is faid t* be in-
fenbed in a right-lined figure,
when every one of the angles of
the inferibed figure touch every
one of the fides of the figure
wherein it is inferibed.

So the triangle DEF is inferibed
in the triangle JSC.

II. In like manner a figure is
faid to be defcribed about a figure,
when every one of the fides of the
figure circumfcribed touch every
om of the angles of the figure about which it is
circumfcribed.

; Si$Z *rian£l<JBC is defcribed about the trian-
gle DEF0 J

5 H P

III. A right-lined figure is {aid to be inferi-
bed in a circle, when all the angles of that fi-
gure which is inferibed do touch the circumfe-
rence of the circle.

IV. ^ right-lined figure is faid to be defcribed
about a circle, when all the fides of the figure

which

77

EUCUDFi Elements.

which is prcumCcribcd touch the periphery of
the circle.

V. After the like manner t circle is faid to
be inferibed in a fight-lined figure, when the
periphery of the circle touches all the fides of
the figure in which it is inferibed.

VI. A circle is faid to be defcribed about a
figure,when the periphery of the circle touches all
the angles of the figure, which it circum-
fcribes.

VIL A right line i«»
faid to be co-apted or ap-
plied in a circle when the
extremes thereof tall upon the
circumference; as the right
line JB.

PROP. L Prohl. i.

In a circle given ABC
to apply a rirht line JB
equal to a right line given »
|D, which doth not ex-
ceed AC the diameter of
the circle.

From the center A
t>y the fpace AE = D
a defcribe a circle meeting with the circle siren a
in B, draw AB. Then is AB b — AE c = D. 3. i.
Winch was to he done. btj.rfrf.n

c confir.

PROP. U. ProhLu

In a circle given
ABC to defenbe *
triangle A B C ,
equiangular to a
C triangle given D.
£F.
Let :the right
line

t a

Digitized by Google

78

a 17. ?.
b 23. 1.

c 32. 3.

d eonftu
C )1. 1.

f

a 13* i«

b 17. 3.

c I}, x.

d 11. 3.

f 13. I.
n 3. ax.

7Ae fourth Bo#k of

line OH <f touch the circle given in A 5 * make
the angle HAC = E, i and the angle GAB =
F. then join BC 5 and the thing is done.

For the angle Bcrs HAC d — E, and the
angle C c = GAB = F 5 e whence alfo the
angle BAC :rr Therefore the triangle BAC
inlcribed in the circle is equiangular to D£F.
Winch was U be done.

PROP. III.

a

Jbout a circle given IABC to defcriht a triangle
LNM equiangular to a triangle given DEP.

Produce the fide EF on both fides ; at the
center I a make an angle AIB =r DEG, and an
angle BIC — DFH. Then in the points A,B, C
let three right lines LN, LM, NM b touch
thedrclft, and the thing is done.

For it's evident that the right lines LN,LM,
MN will meet and make a triangle, 0 becaufe
the angles LAI, LBI are right $ fo that the d
right line AB produced will make the angles
LAB, LBA, lefs than two right angles.

Since, therefore the angle AIB -+ L e = z
'right angles/ = DEG-t-DEF, and AIB g^
DEG ; h therefore is the angle L ■== DEF. By
the like way of argument the angle DEF*
ft therefore alfo the angle N *= D. And there-
fore the triangle LNM defcribed about the cir-
cle i* equiangular to EDF the triangie given.
Which was to be done.

- 1 ■ ■

PRO Pi

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N E17CLIDE'* Element,.
PRO?. IV.

79

J« j triangle given
JBC> to defcnbe a
circle EFG.

a Bifcd the angles
B and C with the a * 1«
right lines BD, CD
meeting in the point
I), b and draw the,
perpendiculars D E, XUU
DF,DG. A circle dc-
fcribed from the center D thro' E, will pafs
thro' G and f, and touch the three lides of the
triangle.

For the angle BDE* — DBF \ and the angle c coxftr.
DEB d-=z DFB ; and the fide DB common, t d iz* ax.
therefore DE DF. By the like argument e 16. i.
DG — DF- The circle therefore defcribed from
the center D pafTes thro' the three points E, F,
G. and whereas the angles at E, F, G are right,
therefore it touches all the fides of the triangle.
Which was to be done.

Schol. '

Hence, The fides of a triangle being known , their J^JKfa-fc,
fegments which are made bv the touchings of tlx cif-
cle inferihed Jball be found, thus ;

Let AB be izy AC 18, BC 16, then is AB +
BC = z8. Out of which fubduft 18 rtz AC -
AE-+-FC, then remains 10 = BE - BF. There-
fore BE, or BF -= $ 5 and confequehtly FC, or

CG, = 11. Wherefore GA, 01 AE, = 7.

s

PROP.

Digitized by

8o

The fourth Book of

PROP. V.

i

\

j About a mangle given ABC to defer ibc a chcli
FJBC.

a i#. and a Bifeft any two fides BA, CA with a perpen-
ti. r. dicular DF, EF meeting in the point F. I fay
this {hall be the center of the circle;
For, let the right lines FA, FB, FC be drawn,
b conftr. Now becaufe At) b -~ DB and the fide DF com-
teonftr.& mon, and the angles FDA c — FDB, therefore is
ii. ax. FB d — FA. After the fame manner is FC ==:
d 4. 1. FA. Therefore a circle described from the cen-
ter F (hall pafs thro" the angles of the triangle
eiven (viz.) B, A, C. Which was to be done-.
w ' Coroll.

* |I« $• * Hence, if a triangle be acute-angled, theceq-
ter fhall fall within the triangle; if right-angled,
in the fide oppofite to the right angle, and if
obtufe-angled, without the triangle.

Schot.

By the fame method may a Circle be defcri-
bed, that fhall pafs thro* three points given, not
being in the fame ft rait line*

PROP*

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EUCLlDE'f Elements.

Si

PROP. VI.

■

In a circle given

EABCD to infcnbe z
fquare ABCD.

a Draw the diameters ant*
AC, BD cutting each
other at right angles in
the center E. Join the
extremes of thefe diame-
ters with the right lines
AB, BC, CD, DA. Aid
^ the tiring is done.

N o\v becaufe the four angles at E are right,the
* arches and c fubtended lines AB, BC, CD, DA b i& u
are equal ; therefore is the figure ABCD equi- c 20. u
lateral, and all the angles in lemicircles, and fo d ; r. 1
d "ght. € Therefore ABCD is a fquare inferi- e io^ Jefi
bed in a circle giveri. {Thick was to be done. t.

PROP. VIL

(4 About
EABCD

a chrU given
to defcribe &
fquare FHIG.

Draw the diameters

the other at right an-
gles } thro' the extremes
• of thefe diameters <*at7*ji
k draw tangents meeting
in F, H, I, G, then I fay it's done.

For becaufe b the angles A and C are right, cb 18. J,
therefore is FG parallel to HI. After the fame c 18. ir
manner is F H parallel to G I. and therefore. FHIG
is a Pgr. and alfo right-angled* It is equilateral
becaufe FG rf = HI ?rrDBe-CAi - FHd^A^u
GI. Wherefore FHIG is a f fquate defcribede i^Jef.u
about the circle given. Which was to be done f z*Aefa<

F Schol. * J

9z

The fourth Book of ,

Schol.

.1

a 7. ax.
b

d 7. /fx.
e 54. 1.

A E j3 A fquare ABCD defended about
7/\\ a circle is di uble of the fquare E F
H r — "/Jl GH inferibed in the fame circle.

For the reftangle HB.rr z
C G £HEF, and HD - *HGF by tke

41.

PROP, VIII.

/?j *r f quart given
JBCD to infcnlt a circle
1EFGH.

Bifed tlie fides of the
fquare in the points
H, E, F, G, cutting one
tlie other in I. A cir-
cle drawn from the
center I through H lhall
be inferibed in the fquare.

For becaufe AH and BF are a equal and h pa-
rallel, c therefore is AB parallel to HF parallel
to DC. After the fame manner is AD parallel
to EG, parallel to BC ; therefore I A, ID," lfef
IC are parallelograms. Therefore AI} d = AE
t = HI = EI ■= Fl IO. The circle there-
fore defcribed from the cepter I thro? H ftall
pafs th?o' H, E, F, G, and tquch the fides of
the fquare, being the angles H, (Jf are-
r^ght. IKhichivastobtdoncJ*

PROP.

Google

EUCLIDE'j Elements:

PROP. IX*

jftofel a /^litfre given
ABCD to iefcribe a
circle EABCD

Draw the diameters
AC, BD cutting one
the other in £. From
the center E thro' A
defciibe a circle, then
I fay that circle is
defcribed about the
f4U3re. „
for the angles ABD and BAC are a half of * 4-
right angles, * therefore EA r= EB. After the It* J*
fame manner is EA r ED - EC. The circle l> 0. *
therefore defcribed from the center E paffes
thro' A, B, C, D the angles of the fquare given.
Wbieh was to be done.

PROP. X.

To make an
tfofcelet triangl*
ABD, having
each angle at the
bafe B and ADB
double to the
remaining angle
A.

Take any right
line AB, and di-
vide it ihC, /iloa ir
that A B x B C
may be equal to

4 2)

ACq. From the center A thro* B, defcribe the
circle ABD ; and in this circle Fatfply BD -r b
AC, and join Al>; I fay ABD is the mangle
required.

F > For

i* 4

84 The fourth Book of

For, draw DC, and thro'the points C, D, A.
c f. 4. t draw a circle. Now bccaufe ABxBC=ACci,*
d $7. ?. it is evident that FD touches the circle ACD
c $ 2. 3 . which CD cutteth ; e therefore is the angle BDC

f I. ax. —Ay and therefore the angle BDC CDA/—
g*z.i. A 1 CDA^-BCD. But BDC j CDA = BDA
h j. 1. h — CBD. k therefore the angle BCD CBD,
k r. ax. and therefore DC l^DE — m AC, n wherefore
16. 1. the angle CDA = A — BDC. therefore ADB
m eon/lr. - lAr ABD. Which was to be done.
n j. 1. This conftiiiftion is Analytically found out
thus ; take the thing for done, and let the right
line DC bifeft the angle BDA ; a therefore DA,
a ?. 6. DB :: CA, CB. alfo becaufe the angle CDA*r=
b conftr. y ADBc r= A, c therefore CA - DC. and be-
c typ. caufe the angle DCB b -A CDA- z A e—
d 6. 1. B, d thence will be DB = DC. /from whence
e 11. t. alfo DB— CA. and fo DA(BA.)CA :: CA.CB. g
i z. ax. whence BA x CB = CAq.
g 17- * Coroll.

Whereas all the angles A, B, D /;make up
h 3*. 1. two right angles, it's evident that A is i of tw#
light angles.

PROP. XL

1 .

/» a circle given ABCDE to defer ibe a Pentagon*
figure ABCDE equilateral and equiangidan
* f o. 4. a Defcribe an Ifofceles triangle FGH, having
b z. 4. eac^ angle at the bafe double to the other 5 b in-

„ I fcribe.

^ 1

Digitized by

EUCLIDE'r Elemtnts. 8;

fcribc a triangle CAD equiangular to the faid
triangle FGH, c Bifecr the angles at the bafeC 9. 1.
ACD and ADC with the right lines DB, CE
meeting with the circumference in B and E.
join the right line CB, BA, AE, ED. Then I
fay it is done.

For it is evident by conftruction that the an-
gles CAD, CDB, BDA, DCE, EC A, are equal;
wherefore the d ai^hes aid e fubtended lines DC, J |#
CB, BA, AE, DE are equal. Therefore the Pen- e 19. 3.
tagone is equilateral, and equiangular / becaule f 27. 3.
the angles of it BAE, AED, Qfc. ftand on equal
£ arches BCDE, ABCD, &c. g L% ax.

A more eafy Pra&ice of this Problem ftiall be
deliver'd at 10, 13.

CcroV.

Hence, each angle of an equilateral and equi- » .

angular Pentagone is equal to f of two right
angles, or -? of one right angle.

ScboL

GeveraUy all figures of odd number of fiies are PctMer^
infaibed in circles by the help of Ifofceles triangles ,
whofe angles at the baft are multiples of thofe at
the top : and figures of even number of fides are in.
fcribei in a circle by the help of Ifofceles triangl$9
whofe angles at tlx bafe are multiples fefquialter of
thofe at the top.

As in the Ifofceles triangle
Jr CAB, if the angle A 22 ? C= . ,

B, then will AH be the fide of
a Heptagone. If A rr 4 C, then
is AB the fide of Enneagone.
But if A c= 1 C, then is AB
the iide of a fauare. And if
A = z ! C CAB will fubtend
the fixth part of a circumference : and like wile
if A = } 1, the* will AB be the fide of Oda-
gone.

(> .

F i PROP.

Tie fourth Book of
PR QP. XUr

About a circle given fJBCLEjo dcfcriU an cqu>
lateia and an equiangular fentaeonc HIKLQ^
» ii, 4, * Infcrib* a pentagon* ABCDE in the circle
given ^ and from the center draw the right lines
fA, FJ$, FC, FD, FE; and to thole lines draw fo

many peipendicularsQAH^HBI^CK^DL^LEG
' * mceane in the points H, I, K,L G. th*:n I [ay it

is oone. For becaufe GA, GE from the fame
hear 16 Vkj'm ^ * touch the circle, <: therefore is GA-
JV ' * Gt, and <f therefore the angle GFA - GFE,
£ j.for.itf, therefore the angle AFE " x GFA. After the
5. 1 'fan** manner is the ancle AFH — FHB, anp!
1 a if coiifeiiiiently the angle AFB =: z AFH. e But
«27 ? tlie angle AFE "r-AFB,/therefore the angle GFA
I - AFH. But alfo the angle FAH * FAG,

s 12 *i *nd thc We FA is common, b therefore HA r=
I X6? AG GE iU&c, feTheiefpre H0,GL,LK,
^ ? " KI,IH the fides of the pentagoue are equal, the
*• V angles alfo are, becaufe double of the equal ai>

gles AGF, AHF, theiefore, Qfc. ... _ .

CW/.

After the fame mantier, if any equilateral and
equiaqgjed figure be defcribed in a circle, and at
the extreme points of the femi-diameters diawB
frqm the center to the angles, be drawn perpen-
dicular, lines to the fajd diameters,! fay that thefp

pe*

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«7

EUCLfDE'x tlments.

perpendiculars (hall make ajK>tht* figure of as
many equal fides and equal angles, dtfcribei
about the circle.

pro*, xni.

_ . i

In an equilatltri
* tnd equiangular pent a-
- ?ow* w J£ CD E,
to infcribe a circlt
FGHK.

a Bifecl two angles a 9, 1,
of the pentagone A
and B with the right
lines AF, BF meeting
in the point F. From
F draw the perpendiculars £G, FH,FI,FK, FL.
Then a circle delcribed from the center F thro'
G will touch all the fides of the pentagone.

Draw FC, FD, FE. Becaufe B A b = BC, and b typ.
the fide BF common, and the angle FBAcrrFBC,c conftr.
^therefore is AF—FC and the angle FA B^FCB, d 4. u
but the angle FAB* rr^BAE^r J BCD. There- e hyf.
fore the angle FCB = ^ BCD. After the fame
manner are all the whole angles C, D, E bife&ed.
Now whereas the angle FGo f ~ FHB, and the f i*. ax*
• angles FBH=FBG and the fide FB is common.^ g x6. 1.
therefore is FG FH. In like manner are all the
right lines FH,FI,FK,FL,FG equaE Therefore
a circle drfcribed from the center F thro'G paffes
thro' the points H,1,K,L and h touches the fides hcor.164*
of the pentagone, becaufe the angles at thofe
points are right. Which war to be done.

CoroH.

Hence, If any two neareft angles of an equila-
teral and equiangular figure be bife£ted,and from
that poiiit in which the lines meet that bileft
the angles be drawn right lines to the remain-
ing angles of the figure, all the angles of the
figure lhal] be bifecled.

I F 4 Schol.

Google

98

•

I

The fourth Bool of

b By the fame method (hall a circle be inferibe^
in any equilateral or equiangular figure.

PROP. XIV,

1 .%

About a pentagone given ABCDE equilateral an%

epnangular to defcribe a circle FABCuE.
Bjfecl any two angles of the pentagone with the

right lines AF, Bf meeting in the point F ;

the circle defcribed from the center F thro* A

(hall be defcribed about the pentagone.
* cor* For let FC, F D, FE be drawn, a Then the an-
b 6. x. gles C, D, E are bifecled > b and therefore FA,

FB, FC, FD, FE are equal ; therefore the cir-
▼ cle defcribed from the center F pafles thro* A,

B,C, D, E all the angles of the pentagone*

Which was to be dem.

SchoL

By the fame art is a circle defcribed about apy
f gure which is equilateral and equiangular.

PROP.

4

Google

NUCLIDE'* Elements.
. PRQP. XV.

«9

In a circle given GJBCD-
EF to info the an Hexagone
(or fix-fidcd jigwc) trilateral
and equiangular J£C l)EF.

Draw the diameter AD ;
from the center D through
the center G defcribe a cir-
cle cutting the circle given
in the points C and E.
Draw the diameters CF,
EP; and join AB,BC,CD,
DE, EF, FA, Then I fay
ifs done

For the angle CGD <t — - -
j of z right a - DGE h - AGF h = AGB.k \< l*
t Therefore BGC - \ 0f z right FGE • ~ 5
therefore the d arches and e fubtenfes AB, Bc\Y%J}A
CD, DE, EF are equal. Therefore the Hexa-e^^
gon« is equilateral, but it is equiangled alfo, fc £ |*
becaufe all the angles of it ftand upon equal *
arches.

mm

Coroll.

1. Hence, The fide of an Hexagone infcribed
in a circle is equal to the femidiameter.

z. Heieby an equilateral triangle ACE may
very eafily be defcribed in a circle given.

ScI)oL

Jo make a true Hexagone upon a right line
given CD.

a Make an equilateral trkngle CGD upon the Jtnilr.
line given CD \ from the center Q thro' C and laeq.
D deTcribe a circle. That circle fhall contain the a 1. 1.
Hexagone made upon the given line CD.

PROP.

_

9* 7h fourth Book of \ A

In a circle given JEBCtoinfcrihea quindeeagonc
a 1 1 m Kte'n'J}?ed fig™*) equilateral and cam angular,
h Va « Imcnbe an equilateral pentagone AfcFCjH
UJl4* in ™ circle given, *nd * alfo an equilateral tri-
angle ABC. then I fay BF is the fide of the

• • quindecagone required.

€ eon tr. For tjle arch AB * is f or Tf of that periphery
whereof AF is * or T therefore the remaining
part dF is of the periphery ; and therefore

gl. , the quindecagone, whole fide is BF, is equilateral :

* *7« but it is equiangular alfo, d becaufe all the angles

lnlift on equal arches of a circle, where every one
14 «>f the whole circumference. Therefore, &c.

Schol

A circle is geo-C 4,8,16,^. by 6, 4, and 9, u
metrically di- )},6, i2,&V. by 15,4, and 9,1.
vided into } 5, 10, 20, gfr. by 1 1 , 4, and p, 1 .
parts. 1 1 *, ?©, 60, grV. by 1 6, 4, and p, 1 .

m Any other way of dividing the circumference
into any parts given, is as yet unknown, where-
fore in the conftruftion of ordinate figures, we
are forced to have recourfe to mechanick artifi-
ces, concerning which you may tonliilt the
Writers of practical Geometry.

# THE

/

THE FIFTH BOOK

O F

EUCLIDE'* ELEMENTS.

Definitions.

Part, i a magnitude of a magni-
tude, a lefb of a gi eater, when the
lefs meafures the greater.

II. Multiple is a greater magni-
tude in refpect of a letter, when
the leffer meafuies the greater.

III. Ratio (or rate) 1* the mutual habitude or
tefpedl of two magnitude.- ot the lame kind each

1 to other, accoroing to ^uantiiy.

In every ratio, that quantity which is referred to
another quantity is called the antecedent of the ratio,
and that to which the other is refer ted is called the
eonfequent of the ratio, as in the ratio of 6 to 4, 6 is
the antecedent, and 4 the eonfequent.

Note, The quantity of any ratio is known by divi-
ding the antecedent by the confequtnt\as the ratio ofiz
to $ is exp efjed by \ , or the quantity of the ratio of A

to Bis g , Whptefon%oftmfor brevity fake, we denote

A C

the quantities of rations thus \ g c% or rr, or -3

that is, tberatio of AtoB is gi renter, equator left 'than
the tatw ofCto D9 And tins note muft be diligently
obferved in the undemanding of the following Book.

Concerning the divers ffecies of ratio's^ you may
fleafe to confult Interpreters.

IV. Proportion i* a fimilitude of ratio's.
That which is here termed Proportion, is Here

rightly called Proportionality or Analogy 5 for

pro-

9*

Digitized by Google

\

** ?be fifth Bool of

proportion commonly denotes no more than the rati*
betwixt two magnitudes.

V« Thofe numbers are faid to haie a ratio be-
twnttthem, which being multiplied may exceed
one the other. j ■

£,ia(A, B,<5.!G,z4. VI. Magnitudes are
*tjo.:C,io.D,ij.iH>6o. (kid to be in the fame
r . - . , fl . »tio, the firft A to the
lccomj U, and the third C to the fourth D.wheo
the equimultiples E and F of the firft A, and the
third C compared with the equimultiples G, H
of the fecond B and the fourth D, according to
*ny multiplication whatfoever, either both to-
gether E, F are lefs than G, H both together,or
-equal taken together, or exceed one the other
together, if thofe be taken E, G and F, H,
which anfwer one to the other.

- SVSVi^lt 5 as A*::C.D.Tbatis9arJ
ytoBjonCto D. which Jignifies that JtoB, andC
teDarewtfjefamcratio. We fometimestlms extrefs
AC 4
q — g rt*t u A.B::C. D.

/aVp " ^^fflaitiidcs that have the fame ratio

twI* A exceeds G the multiple of the fe-
cond B, butf the multiple of the thud C e*-
ceeds notH the multiple of the fourth D, then
the firft A to the fecond B has a greater ratio
than the third C to the fourth D.
■jr A C

H *g C- g , it is not necejfary from this definition

that EJbonld always exceed G> when F is lefs than
M S but it is granted that this vmy be.

Proportionality confifh in three terms at
"rtxreof the fecond fvjplies the flacc of two.
X. When j magnitudes A, B, C are proportio-
ns

Digitized by Godgle

EUCLIDFi Elements.

nal,the firft A fliallhave a duplicate ratio to the
third C of that it has to the fecand B : But when
4 magnitudes A,B,C,D are proportional, the firft
A fliallhave a triplicate ratio to the fourth D of
what it had to the fecoudBj and fo always in or-
der one morels the proportion fhall be extended

K A

Duplicate ratio is tlms exprejfed ^ = *g twice, that

fx, the ratio of A to C is double of tlx ratio of J to B.

A A

Triple ratio is thus exprejfed ; ^ thrice. That is

the ratio of A to D is triple of the ratio of A to B.

•ii denotes continual propertionals ; as A,B,C,D ;
er 2, 6, 18, 64, arcZ.

XL Magnitudes of a like ratio, are antecedents
to antecedents, and confequents to confequenu 5
As if A.B :: CD. A ayii C * and B and D arc homo*
logons or magnitudes of a like ratio.

XII. Alternate proportion is the comparing of
antecedent to antecedent, and coniequent ta
confequent. As if A. B :: C. D. therefore alternMely%
er by permutation, A.C:: B. D. by the itf . of $.

In this definition, and the 5 folkwing, names are

med in their Explications.

XIII. Inverfe ratio is when the confequent is
taken as the antecedent, and fo compared to the
antecedent as the confequent j As A. B : - C. D.
therefore generally B. A :: D. C. by cor. 4. 5.

XIV . Compounded ratio is when the antecedent
and confequent taken both as one are compared
to the confequent it felf. As A. B :: C. D. there-
fore by compojition A B. B :: C -*> D. D by 18. J.

XV. Divided ratio is when the excels wherein
the antecedent exceeds the confequent, is com-
pared to the confequent. As A.B:: CD. therefore by
divifioji A-B. B :: C— D. D. by 17.

XVI.

Tie fifth Book of

XVI. Converfe ratio is when the antecedent is*
compared to the excels wherein the antecedent
exceeds the consequent. As A. B :: C. D. there-
fore hy converfe ratio A. A B :: C.C D* by
the corolLofthe 19. of the 5.

XVII. Proportion of equality is -where there
are taken more magnitudes than two iu one ol-
der, and alio as many magnitudes in another or-
der, comparing two to two being in the lame ra-
tio ; it comes to pafs that as in the firft o derof
magnitudes, the fitft is to the laft, fo in the fe-
cond order of magnitude^ is the fit ft 10 the laft.
Or otherwife : it is a companion or the extremes
together,the mean magnitudes being taken away*

XVIII. Ordinate piopovtionality is, when as
the antecedent is to the confequent, fo is the an-
tecedent to the confequent, and as the confe-
quent is to any other, fo is the confequent to
any other. As A.B :: D. E. alfo B.C :: E.F. itJbaU,
Mjc tnte alfo A. C :: D. F. by the zz. of the •

XIX. Inordinate proportion is, when three
magnitudes being put, and others alfo, which are
equal to thefe in multitude, as in the firft mag-
nitudes the antecedent is to the confequent, fo in
the fecond magnitude is the antecedent to the
confequent , and as in the firft magnitudes the
confequent is to any other, fo in the lecond mag*
nitudes any other thing to the antecedent. As
A. B :: F. G. alfo B.C :: EF. itjball be truein inor-
dinate proportion. A. C :: E.G. by the z$. of the $.

XX. Any number of magnitudes being put ;
the proportion of the firft to the laft is com-
pounded out of the p»oportions of the firft to
the fecond, the fecond to the third, and the
th rd to the fourth, andfo forwards till the pro-
portion aiife.

Let there be any number of magnitude^ A, B,C>

D. by this definition ~ = £ -\- 7? +■ ^

U sj w U .

Axiom;

%

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Axiom*.

j|fognitude£ equimultiples to the Tame multi-
ple, art alio equimultiples betwixt thepjfelves.

Jf there be a number of magnitudes how many fo-
tver, JB, CD equimultiples to a like number of mag-
nitudes £, F each to other ; how multiple one magni-
tude JB is one Eyfo multiples a)e all the magnitudes
JB Cp to all the other magnitudes E f.

Let AO, GH, HB the parts of the quantity
AB, be eaual to E, and alfo let CI, IK, KD the
parts of tne quantity CD be equal to F. The num-
ber of thefe are put equal to thofe. Now whereas
AG^CU— E+tf; jandGH + lK— E j hha
and HB-+ KD:rr E-+ F; it is evident that AB-»-
CD does fo often contain E^1- F as one AB con-
tains E. Which was to be done.

PROP. II,

If the firjl JB be equimultiple la
the fecond C, as the thpd DE is to
the fourth F, and if the fifth BG be

a 2, etXi

Ht

E

:i ii

equimultiple to the fecond C as the
fixth EHis to the fourth F $ then fbalL
the firft compounded with tlx fifth
(JG) be equimultiple to the fecond
C, at the third compounded with tlx
fixth (DM) is to the fourth F.
_ ^ The number of parts in AB e-

AC DE °*ua* eac^ to P *s Pu* equal to the
number of parts in DE. whereof

each pan is equal to F. Likewife the number of

parts in BG is equal to the number of parts

in EH. Therefore the number of parts in AB

•fBO U equal to the number of parts in DE

EH*

Digitized by Google

96

a 2, ax.

a hyp,

b 20 j.

The fifth Book of

FH. 4 That is, the whole line AG is as equt*

multiple of C, as the whole line Dfl is of
i^Wf h was to be dem.

eauimultiple of
Therefore EG h
cond B, as FK -
the fame way of
multiple of B,
Winch mi to fa

PROP. III.

& H

Ifthefirft A be equimultiple
of the fecond 2?, and the third C
of the fourth D, and there be ta-
ken Ely FM equimultiples of the
firft and t1mdy then mil each of
the magnitudes taken be alike
equimultiple of both, the one El
to the fecond t, the other FM to
the fourth D.

Let EG, GH, HI the pans
of the multiple EI be equal to

A, alfo let FK, KL, LM the
parts of the multiple FM be
equal to F, a the number #f
thefe is equal to the number
of thofe. Moreover A (that
is) EG or GH, or HI is put as

B, as C, or FK, gft.of D. *
h GH is equimultiple of the fe-
t KL is of the fourth D. c By
argument is EI (EH i HI) as

as FM (FL h* LM) is of V.

done.

VI

• > •

>mt ■ - " —

• 1

» »

I . It
t

i: • i

*ROP.

• : i • :
•1*

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EUCLIDE'i Elements:

f7

IEABCI

mi

I

PROP. IV,

Ifthefirjl A have the fame ratio
to the fecond By as the h'nd C to the
fourth D j then alfo £. and F the
equimultiples of the firft Akand the
third C, Jball have the fame ratio to
G and H the eauimuitiples of the
fecond B and the fourth t), according
to any multiplication, if fo taken as
they anfwcv each to other (£. G ::
F. H.)

Take I and K the equimulti-
ples of E and F } and alio L and
M the equimultiples of d and H.
a Then is 1 as multiple of A* as a 5.
K of C 5 a and alfo L is as mul-
tiple of B, as M ot D. Theie- ^
fore whereas it is A. B :: CD;

according to the iixth definition,
if I be cr, =, -d L, then confe-
quently after the fame manner is
K rr, ~, "3, M. Therefore when
I and K are taken as multiples of
E and F, as L and M of G, and
H, then will it be by the feventh definition E.
G F. H. Which was to be dem.

CorolL

From hence is wont to he demonjlrated the proof
of inverfe ratio*

For becaufe A. B :: C. D, therefore if E rr,
-3 G, then is c like wife F cr, =, ^ c i*def f4
therefore it is evident that if G cr, =, ~3 E,
then is H cr, rr, -3 F } d therefore B. A ;: D. C. d 6. def f «
Which was to he dew,

't

G PROP-.

• 4 • • •«

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Tie fifth Book of

PROP. V.

£ — r

_j Jf a m&nitudc AB
E he as miHtiple of a
magnitude " CD, as - a
part taken from the one

b 6. ax.
e 3. ax.

AE of a fart taken from the other Cf\ the refidue
of the one fiall be as multiple of the refidue o f tfje
other FD as the whole AB is of the whole CD.

Take any other GA, which fhall be as mul-
tiple to FD the refidue, as AB is of the whole
CD, or as the part taken away AE is of the part
taken away CF. a Therefore the whole GA -f-
AE is as multiple of the whole CF ^ FD, as
the one AE is of the one CF, that is as AB
is ot CD. therefore GE h — AB \ and c fo AE
that was common being taken avray, there re-
mains G A — EB.

PROP. VI ! !

If two magnitudes ABJCD he equi-
multiples of two magnitudes is, F ;
and fome magnitudes AQ and CH t-
qiiimuluples of the fame £, F, be ta-
ken away ; then the rejidues GBy HD
arc either equal to thoje magnitudes
£\ F, or elje equimultiples of them.

For becaufe the number of pans
in AB, w hereof each is equal to E,
is .put equal to the number of parts
in CD, whereof each is equal to
F, and alio the number of parts in
AG equal to the number of parts
in CH i If from one you take AG, and from the
a j. ax.i^ other CH, a ttfen lemains the number of pans
in the remainder GB equal to the number of
parts in HD. therefore if GB be once E, then is
HD once F. if GB be many times F* then is HD
lc of F. Which was to he ucvi.

TB

G

\

PROP

f

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EUCLIDE'i Eiemcntsi

99

S

PROP. VII.

Equal magni-
tudes A and B
have to the fame
magnitude C the
fame proportion or ratio. And one and the fun*
magnitude C has the fame ratio to equal magnitudes
A and B.

Take D and E equimultiples of the equal
magnitudes A and B, and F any- wife multiple
of C ; then is D a - E. Wherefore if D r,r:,a6»^
-3 F, then alfo E will be cr, ~3 F. b there- b 6. def. $.
fore A. C :: B. C. and c by inverfion C. A cC.c cor. 4. $.
B. Which was to he dem.

Scbol.

If inftead of the multiple f, two equimultiples
be taken, it fliall be the fame way provM that
equal magnitudes have the fame ratio to ather
magnitudes that are equal between themfel ves.

PROP. VIII.

g Of uneaual magnitudes ABy AC, the
greater AB has a greater ratio to the
q f4me third line Dy than thelejfer AC\ and
m . the fame third line D bath a greater ratio
to the lejfer AC, than to the greater AB.
Take EF, EG equimultiples of the
t fakl AB, AC,fo that EH being multi-
ADple of D be greater than EG, but leiler
than EF. (which will eafily happen, if
both EG and GF be taken greater than
D.) It is manifcft from 8. def 5. that
A3 AC

t n he icm.

D

D

AK ^ AC

JPhich

Gz

PROP

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the fifth Book of

PROP. IX.

\ Magnitudes which to one and the fame mag-
' nitude have the fame ratio^ arc equal the one
to tlx other. And if a magnitude bane the
fame ratio to other magnitude s^thofe magni-
tudes are equal one to the other.
ABC « Hyp. If A.C::B.C; I fay that A— B,
For let Abe jgeateror lefsthan C.athenis

^ C or -71 Winch is contrary to the

Hypothefis.

z. Hyp. If C. B :: C. A. I fay that A=:B. For

C C

let A be cr B, h then g tr Winch is againjt
the Hypothefis.

• »

PROt. X.

Of magnitudes having ratio to the fante
magnitudcx that which has the greater ra-
tioy is the greater magnitude \ and that mag-
nitude to which the fame canies a gteatev

J f t ratio is the lejfer magnitude. Ej

ABC ^.KiiS I faythatAc-B

For if it be faid that A~B, a then A.jC:: B.C.
Which is contrary to the Hyp. If A"3B,fi then is
A B

~3 ^ JFZfflS M j//b againjt the Hyp. £ t

2. %>. If 2 c- H I fay that B-2A. for if you

fa jr B— A, it's againft the Hypothefis, for itu§ll
* follow that C. B:; C. A. If you fay B c A,d

C C

then i> ~ zm »g# iWnch is alfo againjt the Hypt

PROP.

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EUCLIDE'/ Elements.

Proportions wMcb are one ana the fame to any
third, are alfo the fame one to another.

Let A. B :: E. F, and C. D :: E. F. I fay that
A.B :: CD. Take G, H, I the equimultiples of
A, C, E ; and K, L, M the equimultiples of B,
D, F. Now a becaufe A.B:: E. F, if G cr, rr, a / *

b then after the fame manner I cr, — »bo\i*/.*.
-3 M. And likewife a becaufe E. F :: C. D. if
I c% — , -3 M, b then is H likewife cr,—, ~3 L.
c wherefore A, B :: C. D. Which was to be Jem. c 6.dcf. J.

SchoL

Proportions that are one and the fame to the
fame proportions, are the fame betwixt them-
felves.

PROP. XII.

G H 1

A C E

B D — i F

K ~ L M

If any number of magnitudes Afi \ C,D \ EandF
be proportio7iais \ as one of the Antecedents A is to ont
of the Confequents B ; fo are all the Antecedents A,
C, E to at the Confequents B, D, F.

Take the equimultiples of the antecedents G,
H, I, and of the confequents K, L, M. Becaufe
that as multiple as oneGisof one Aya fo multi-a 1. $.
pies are all G,H,I,of all A,C,E ; and likewife
as multiple as one K is of one B, fo multiples
are all K,L,M,of all B,D,F. moreover becaufe A. 7
B b :: C. D b :: E. F. if G be tr, rx ,or"?K. then b bjf.
will H likewife be r-,^,^L,andICV~,-^ M.
and fo if G ry-^K. in like manner will G-+-
H-+-I be cr,— c wherefoie A%B :: c 0. def%.
A C~* E. B -+ D F. Which was to be dem.

G }■ Co-

Digitized by Google

icm the ffth Book t>f

Coroll.

From hence, if like proportionals be added to
like proportionals, the wholes lhall be pror
portional.

prop. xra.

• H — —J

C E

D F

— L— . M>

Ifthefirft Ahave tie fame ratio to the feeoni Bf
that the third C has to the fourth D ; and if the

Tajee £}, H, I equimultiples of A, C, E, and

£ „ K»L> M, equimultiples of B, D, F. Now becaufc

» ^ that A. B a C, D. if H c- L. * then is Q cr K.

C E

b SJff.s.hn becaufe g er i it may be that H c

A 1>

c 8. <fc/.s.L, and yet I not cr M. e Therefore g qr jr .
Which was to he dem.

* '

if g -5 1, then alfo is | -3 |, Alfo, if

trBC" f»then is I c-|. And if

4 ~ C E , . A E
£ -3 -^ then u S-3S.

PROP.

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EUCUDE'j Ehmentu

PROP. XIV.

I If the fir ft A have the Jam* ratio to the
T « • fteond B9 that the third C has to the fourth
1 D ; and if the firft A I e greater than the
j third C ; tlxn JhaU the fecond B be greater

1} than the fourth D. But if the firft A be
i 1 equal to the third C, then thefecondBfbaU
be equal to the fourth D. but if A be lejfer,
the is B alfo letter.

AC

Let A cr C a then >- cr ^ b but a 8. 5.
A BCD * * hhyf.

— . ^. c therefore <: theref.c 13. 5.

B D D B

B cr D. By the like way of argument, if A~3 C,
d then is B ~D D. But if A be put equal to C, d 10. {.
then C. B :: e A. B /:: C. D. g therefore B ^ D. e 7. 5.
#7>i<;fi ro dem. - f fy/.

A C °
By an argument a fortiori, if *g "3 |y an<* A 9# **

c C. then is B cr D. Likewife if A B, then
is B = D. and if A cr, or -3 B, then alfo is C

cr or D.

* : ■ 1. • . ,

PROP. fXV. •

Tarts C and F are in the fame ratio^with
their like multiples AB and DEy if taken
correjbondently. {AB. BE :: C. P.)
xr Let AG, GB parts of the multiple
AB be equal to C 3 and let DH, HE
parts of the multiple DE be equal to F.
a The number of thefe parts is equal to* hyf.
the number of thofe. Therefore whereas 5 7. J#
-±lb AG, C :: £)H. F, and GB. C :: HE. F,c Ilt 5.
therefore is, AG- GB (AB.) DH-HE
(DE) C. F. Which vas to be dem.

G 4 PROP.

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IP4

rh

Booi of

I ■ ■ i

fifth
PROP. XVL
1(3- 1 T

e

4

If four magnitudes J, By C, D he Proportionaljhey
alfo fball he alternately proportional (J. C :: B. D.)

Take E and F equimultiples of A andB; take
alfo G and H equimultiples of Cand D. There-
? f $. 5. fore E. F a :: A. B h :: C. D a :: G. H. Where-
b hyp. fore if Ecr, "D G, <: then likewife is F c,
c rr.5.gf -:■ H. d Therefore A. C :: B. D. Which was

14

d 6. def.f,

to bp dem.

Sehol

pi. h

e <5. def.

Alternate ratio has place only then when the
quantities are of the fame kind. For heteroge-
neous quantities are not compared together,

PROP. XVII- /

N

1

-HE

C T*

a f

If magnitudes compounded he
proportional JB. CB :: DE. FEJ)
they fball be proportional alfo when
divided (AC. CB :: DF. FE.

JakeGH, HL, IK, KM, in
order the equimultiples of AC,
CB, DF, FE 5 and alfo LN,
MO, the equimultiples of CB,
: ' FE. The whole GL is a as
multiple of the whole AB, as
. pne GH of one AC, h that is
as IK of t)F, c or as the whole
IM of' the whole DE. Alfo
I H^(HL^JLN) is as ^multiple
% - of CB, as KO (KM -+MO) is
E of FE. Therefore, whereas by
Hyp. AB. BC :: DE. EF. if GL
becr,^ r^HN, then likewife
e will 1M c-,— ,-jKO. Take
I from thefe HL, KM that are
' equal i

; Digitized by Google

equal ; and if the remainder GH be c, ~3'
LM, / then will IK cr. — f ~ MO, g whence f $. ax.
AC. CB ; ; DF. EF. ™ s to be dem. g 6. def. <„

PROP. XVIII.
F If magnitudes divided be proportional
C (AB.BC :: DE- EF) the fame alfo

I _ r\ being compounded Jball be proportional
J, | ^ (^C. CB :: DF. FE.

i E For if it can be, let AB. CB.: DF. <
B FG -3 FE. a Then by divifion will a 17. j.

AB. BC .v DG. GF. b that is, I)G, b byp. &
GF i: DE. EF. and being DG cr ir. 5.
AD DE, c therefore is GF ~EF. d Which c 14. %.
is abfurd. The like abfurdity will follow if it d 9. ax*
be faid AB. CB :: DE. GF cr F£.

PROP, XIX.

C // the whole AB b§

A J B#: to the whole DE as the v

F E /tf rf taken away AC is

D — [- /a part taken away

DFj then JbaU the reji-
iue CB be to the rejidue FE as the whole AB is to
the whole DE.

Becaufe a AB. DE ;; AC. DF, b therefore by a byp.
permutation AB. AC ;; DE. DF. c and thence b 10. 5.
py divifion AC. CB DF. FE. b wherefore c 17. *•
again by ptrrputation AC. DF CB. FE. d that d kit* &
is, AB. DE .v CB., FE. Which was to be dem. $•

Coroll,

Hence, If like proportionals be fubftra&ed
from like proportionals, the reiidues lhall be
proportional.

*. Hence is converfe ratio demonjl rated.

Let AB. CB ;: DE. FE. I fay that AB. AC A
DE. DF. For by a permutation AB. DE CB. a Io\ ^
FE. b theref. AB. DE AC. DF. whence again 5 17, 5,
by permutation AB. AC ;: DE. DF. Which was
to be dem. •% •

PROP.

Digitized

I

II

•v.v

Aii

10* Tbt fifth Bool of

If there be three magnitudes JJI,C9
X and others D, E, F equal to thofe in
^number, which being taken two and
two in each order are iu the fame ra-
tio, ( A B :: D.£; andB.C:: E.Fy)
and if of equality the firjtj he greater
than the thxdC\ then Jb all the fourth
D he greater than the fixth P. But if
A B C D E F the firft J be equal to the thirdCjbcn

the fourth D is fo to the fixth F\ and
if J be lefs than C,fo D is Ms than F.
a hyf. ? i. Hyp. Let A~ C. Becaufe a E.F :: B. C. by b in-
bcor4*i. C A

c hyp. and verfion fhall be F,E C.B. c But «g ~3 -^therefore

8* *• FAD

d/ri#iij.g-3g or-5 e therefore DrF. W.W.tobedem.

* 2. Hyp. By the fame way of argument, if A-?C,
e - t it will appear that D "3 F.

S -J DJ6.f therefore is D F, Whuhwaitobedem.
9 h * PRO P. XXL

If there be three magnitudes A,B£,
and others alfo D, is, F equal to them
in number, which taken two and two
are in the fame ratio i and their pro-
J portion inordinate ( A. B :: E. F.
I and B. C :: D.E.) and if of equality,
| I j the firft J be greater than the third
' " I then is the fourth D greater than
A B C D E F the fixth F : but if the firft be equal
i to t%e third, then is the fourth equal

to the fixth i if lefs, fo is the other likewife.
U Hyp. If A cr C ; then became a D.E :: B. C.

C A

therefore inverfely E.D C.B.but ^~37T cthm*
JH E A . . E

cfcholl

d U z. Hyp.

b 8. J.

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ZVCUDE's Elements. k>7
%. Hyp. By the lflte arguments, if A-3C,tfcen

3. ff/p. If A = C ; then becaufe £. D :: e C.e 7.
B .v e A. B / E. F, £ therefore is D = F. f /yp.
JTAif J was to be dem. g 9» S»

prop. xxn, f

«

1

1

?ll
I

I. I If there be any mm-

I I ber of magnitudes J JSf

I I C, /ut4 o/W e^iia/ 0

*fom in number Jl>,£,F,
tafow two and

£ * 9 N 5 I £ ° tio {A. B :: 6. EL and >

HKM j. c .v E.F) they /ball
be in the fame ratio al-
fo by equality, (A. Ci:
D.F.

Take G. H equi-
multiples of A, D ;
andI,KofB,E;and
alfoL,M ofC,F.

Becaufe a A. B .v a hyf.
D. E, b therefore G. b 4. t.
I .v H.Xw attd in like ' *

manner L L K. M«

therefore, if G cr,^r, • J
"TJ, L, <r then is H cr, t=r, ~3 M. d therefore A. c zo. f.
G .•: D. F. By the fame way of demonifratjon if & 6» «fe/.J.
further C. N F.O, then by equality A. N .7

D. O. Which was to be dm* _

1 * * ' »

1

I

■

PROP.

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1

v '1

A

G

B C
H K

a 15. %

b hyf.

c 4. J-

■

1

I

I

PROP. XXIIL

If there be three magnitudes
AJ&fii and others D, E> F, e-
^tt/i/ to them in number ,wJ)icb
taken two and two are in the
fame ratio, and their propor~
D E F twnality inordinate (J.B :: E.
I L MF. and B.C:: D. E.) they JbalL
be in the fame ratio alfo by
equality.

Take G, H, I equimiilti-

$les of A,B>Djand alfoK,L,
4 equimultiples of C, E,
f ThcnG.H::/x A.B i E.
f a :;L.M. Moreover becau
j I b B. C :: D.E, thence is c H.

I j K :; I.L; therefore G,H,K,

and I,L,M are according to
2r.5. Wherefore if G be c%
=*, ts K, then is likewife Icr, ~ , iM. and

, - fo i confequently A. C ;; D. F. Which was to be
d o.aej.$^emonjiraU(i9

If there Ve more magnitudes than three, this way
of demonftvation holds good in them alfo.
_ . Coroll.
*tz.&z%. From hence * it follows that ratio's com-
$. and zc. pounded of the fame ratio's are among them-
dcf. u ielves the fame 5 as alfo the fame parts of the
. fame ratio's, are among themfelves the fame.

PROP. XXIV.
A — — - — B- — G If the firjl viagnitude JB
C — — — have the fame ratio to the fe~

D E H cond C, which the third D E

F has to the fourth F \ and if

the fifth BG have the fame
ratio to the fecond C, which the fixth EH has te the
fourth Fy then JbaU the firjt compounded with the
fifth (ACfy have the fame ratio to the fecond C,
which the third compounded with the fixth (DH) has
to the fourth F.

For

V

Digitized by Google

B

EUCLIDE'i Elements'.

For becanfe a AB.C ;; DE. F, and by the Hyp. a hyp
and inverfion C. BG :: F. EH 5 therefore by I e- b zz.V
quality AB. BG DE. EH. whence by com-
pounding AG. BG :: DH. EH. Alfo c BG. C ;: c jfo

EH. F. Therefore again by & equality AG. C
DH. F. Wbkh was to he devi.

PROP. XXV.
If four magnitudes be proportional {AB.
CD :: E. F) the greateft AB and the lenjt
F Jball be greater than the reft CD, and E.

Make AG=E, and CH^F. Becaule
AB. CD a E. F :: b AG.CH. c thence * hyp.
is AB. CD GB.HD. d but ABctCD. b 7- 5-
e therefore GB cr HD. But AG -+ Fr= C 19. $•
E * CH, therefore AG -4- F -1 GB c~ d
pE CH • I HD,that is, AB -+FrE efchohi^
CD. IP/w/j W7ji *0 dem. %•
Thefe Proportions which follow are not Euclide'^
hit taken out of other Authors, and here fubjoinek
becaufe of their frequent ufe.

PROP. XXVI

If the fift Lave a \
greater propotion to
the fecond, than the
third to the fou»th%
then contrarywife, by
convetjion, the fecond Jball have a left proportion to
tbefirft, than the fourth to the thud.
t A C B D

1rD' 1 fay that T ~3 7> Forconcrive
c — E a E

B — B' * thercfore Fff h whence AcrE.*there- * * 1* 5-
f . i or - jpfaa Wtff to k ^ j
PROP. XXVEb . • •

if tbefirft have a gi eater pro-
portion to the fecond, /ban tbe
third to tbe fourth : then alter"
• " natelytbefirjljhallbaveagraiter
fopormn to tbe tbird,tban tbe fetond to the fourth.

Let

A

E

C
D-

-

fore -*
A

A

B
£.

C
D

* r.
4

Digitized by Google

j io The fifth Book of

A C A B

Let «g cr g. then I fay ~ c g. For conceive

EC A E

a 10. $. B = B" therefore A^E,iand therefore ^«r~

b i. ^ B

c 16. J. * or|y JPwi *w to be dem.

prop, xxviii. ' q

if /7;e J grt&tir proportion to the fecond

• - thin the third to the fourth, then thefirjl compound-
fc . ^ jvir/; fecond Jh all have a greater proportion to
: the fecond, than the third compounded with the fourth
to the fourth.

T AB DET, , AC DE .p „
LetE^ W Uaythat BC^EF' ForC°n-

a 10. s- ceive 5? =Si- * therefore is ABcrGB. add BC
c & 5. " to eac^ Part« t*ien * ^ T c tllere*orc

d I& ec f§- J thal is if* 7^ * *f dem*

PROP. XXIX.

If the firfi compounded with the feeond have a.
greater proportion to the fecond, than the third com^
founded with the fourth Mi to the fourth ^ then by
4ivifion the firjl JhaU have a greater proportion to the
fecondj than the third to the fourth.

T AC DF , _ r AB DE Votcan
s . , ^BC^EF then I fay bc^T^. Forces

a 1* 5- ceive =?5. * ttwrfACcrGC. Takeaway

c 8* * * BC, that is common , then > remains ABcrGB; c
v °* t* * AB OB DE
t7* J* Therefore cr g c i or . Vhich vat to he

tep,. : V ; f ' PROP.

Digitized by Google

EUGUDE* Elmmi.
'• PROP. XXI. ' "v ^

a : B If tht firfi

A ———I—!-! C founded with the ft.

D~ 1 — F eond hit* agrutt*

tit

w t -

con<}9 than the third compounded with the fourth has

to the fourth , then fa converfe ratio JbaU the fit ft

compounded with t%4 fecond have a leffer ratio to tht

firjfy than the third compounded with the fourth JbaU

have to the third.

AC / DF T r AC DF

Let r- ^ Ttai IMay that^ m

For bccaufc that g^a tr i therefore by di- a hyf.

AB DE k r b * f -

g£ tr by converfion * therefore c 26. 5,

BC EF dz8. 5.

Ag"3 DE and J therefore by ounpofitkm

A5 '^81. WUchwastohe

• £>£•

PROP. XXXI.
A D If there he three msg~

C F — — a//b D, £, F/^flf

G ^ /<? ffem t n nunile* $ ami

H— if there he a gtvita

jrofortion of the frjl of the former to thefecowL
than time is ofthepft of the laft to tlseir fasti
/A D \

{ g-c- f^J and there lealfo a \ great** frofortknef

the fecond of the fivft magnitudeelo thethirdjbcm there
is of the^econd of the laft magnitude to then thirl

[ ^ C" j}, J Tfjen by eptalitj aifo fbalt tlx 4

thefoftefthf former fiiagnitvdes to tht third Je
greater than the ratio tie fijl of the fattfr mag-
nitudes to the third (c^v|?,^

Owa-

Digitized b

, *fc fifii . Book of

Conceive ^ — ^« therefore is EcG, and J

A A . H
therefore g r |i Agaih conceive g

H A , H A

c therefore »g, therefore much more g g«

AH

J wherefore A cr H, e and confequemly £ c* £
D

PROP. XXXII.

A * — «« D — ■ If there he thee mag-

B ■ E ' ™tuiei A and 0-

q r p fc. /few D, J?, F, ef kj/ to

^ innurttba'i and

^* 1 "u *fore #e j greater pro-

\\ portion ofwefirjtoftbe

formkr magnitudes to the fecond, than there is of the

fecond of the latter to the third (g c*

ratio of the fecond of the former to the third he
greater than the ratio of thefirjl of the latter to the

fecond (g-c- 5# ) then by equality alfoflattthe

proportion of the firfi of the former to the third, he
greater than that of the firjl of the latter to the third
/A D \

The demonftration of this Proposition is alto-
gether like that of the precedent. ;

PROP. XXXIII.
E , If the proportionof the whole AB

A I — B to the whole CD he greater than

Q t— D the proportion of the partjaken a-

J? way AE to the part taken away

CF\ tben jbatt alfothe ratio of the
remaindes EB to the remainder FD he greater than
that of the whole AB totUyphofe CD.

pe-

Digitized by Google

EUCLTDE'/ Elements.
AB AE

BecaufethatgpflCTgp. J therefore bypermu-
AB CD

tatian— cr g-p, ^ therefore by converfe rgtio

AB CD . AB EB

£g~3p|3> and by permutation again ttf^fg.

PROP. XXXIV. 1

A . D • — If there be any

B— — E >/ number of magni-

C— E^— - tude s, and other sdL
G ■ ■ ft — X- ~ / fo equal to them in

t m ^ . Jkl number ; a wrf f £#
proportion of the fit ft of the former to the fir ft of the
latter y be greater than that of the fee and to the fe+
cond, and that greater than the proportion of thi
third to the third, and fo forward: all the former
magnitudes togetlier JkaH have a eater ratio to all
the latter together, than all the former, leaving out
thejlrfty Jball have to all the latter, leaving out the,
firfti but lefs than that of the fir ft of the former to
the firft of the latter \ and laftly greater than that
of the.laft of the former to the la ft of the latter.

You may pleale to conlult interpreters for the
demonftration hereof, we having for brevity
fake omitted it, and becaufe 'as of no ufe in
thefe Elements*

The End of the fifth Book.

H ' TH5

4

THE SIXTH BOOK

O F

EUCLIDE'i ELEMENTS:

Definitions .

Ike right-lined figtirtsf ABC,DCE)
are fuch whofe ieveral angles are
equal one to the other, and alfo
their fides about the equal angles,
proportional,
the apgle B - t)CE, and JB. BC :: DC. €£.
AIJotheanglejhzDJandBJ.JC::CD.DE. Ufib
the angle ACB — E, and BC. CJ :: CE. ED.

tt. Reciprocal figures arc
(BD, BF) when in either fi-
gure are the terms antece-
Gr dents and consequents of ra-
tio's, that is, AB.BG :: EB.
BC.)

III. A right line A3 is
faid to be cut according to
C Bniean and extreme proper-
tion, when as the whole AB
is to the greater fegment AC, fo is the greater
lament AC to the lefs CB (AB. AC :: AC.

IV. The

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EUCLIDE'j Element s. uy

IV. The altitude of any fi-

f;ure ABC, is a perpendicular
ine AD drawn from the top A
to the bafe BC.

V. A ratio is faid to be compounded of two ra-
tio's, when the quantities of the ratio's being
multiplied the one into the other, do produce any
ratio. As the ratio of J to C is compounded of the rations

A B A, AB zxojef.f*
of J to B andBtoC. For g+ g«=£*=£c« b 15.4,

PROP. I.

Triangles JBC, ACD%
and parallelograms BC-
jEy CDF J which havt
the fame height, are in
proportion one to the
other, of their hafesBC,
CD are.

a Take as many as you pleafe, BG,GH,eaual
to BC, and alfo Dl sg CD. and join AG, AH," *•
AI.

b The triangles ACB, ABG, AGH are equal, b j& u
and* alio the triangle ACD ~= ADI. There-
fore the triangle ACH is as multiple of the
triangle ACB, as the bafe HC is of the bafe
BC \ and the triangle ACI as rtoultiple of the
triangle ACD, as tne bale CI is of CD. But
if HC - ~, -a CI, c then is likewife the tri-cMj8.r.
angle AHC cr, —% ~3 ACI * and d therefore d o.dtf.j.
BC. CD :: the triangle ABC ACD t Pgr. CE. e 4*««- &
CF. Which mas to be dem. ****

Scho-

4 • •

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1*6

the fzxth Book of
Scholium.

a 1. i.
c 1.6.

B K C 1XE MS

Hence, TrUngks ABC, DEF, and Pgrs. AGBC,
DEFH, whofe ki/csBC, EPare equal, are infuch
proportion as their altitudes, AIyDK are.

a Jake IL = CB, and KM rr EF ; and join
LA, LG, AID, MH. then is it evident that the
triangles ABC.DEF ;: b ALI.DKM :: c Al.DK ::
. AGBC. DEFH. Which was to be dew.

><j r

a 17.1.

b 7. 5.
c 6.
du.

c 1. 6.
f 19. 5.
8 S9- 1.

PROP. IL

If to one fide BC of a triangle
ABC be drawn a parallel right line
VEy the fame foall cut the fides of
the triavgle proper t ionaHy( AD. MJ)
:: AE. EC.) And if tire fidei of the
^triangle be proportionally cut (AD.
BD :: AE. EC) then a right lint
DE joined at the fedions D, EJjaU
le parallel to the remaining fide of the trimgle BC*
Draw CD and BE. "

1. Hyp. Becaufe the triangle DEB a ~ DEC,
I therefore fhall be the triangle ADE. DBE
ADE, ECD. But the triangle ADE. DBE :: t
AD. DB, and the triangle ADE. DEC: AE;
EC, d therefore AD. DB AE. EC.

2. Hyp. Becaufe A D. DB AE. EC, * that is
the triangle AD£ DEE ADE. ECD 5 / there-
fore is the triangle DBE =s ECD 5 and^ there-
fore DE,BC are parallels. Which was to be dem.

Scholium.

If there be drawn many parallels to one fid©
of any triangle, then all the fegmcutsgf the fides

flj aJ^

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EUCLIDE'j Ehmtntt.

§

fhall be proportional ; as is cafily dcducibl*
from the precedent.

ii7

PROP. III.

If an avgle BAC of a trian*
gle BAC be bifeBed, and the
right line JD, that bifeds the
angle, cut the baje alfo ; then
Jball the fegments of the bale
have the fame ratio that tie
other fides of tlye triangle have
(BD. DC:: AB. Ac) And if the fegments of the
bafe have the fame ratio, that the other fides of the

triangle have {BD.DC:: AB. AC.) then a right
line AD drawn from the top A to the fef
bifett that angle BAC of the triangle,

a riP

all

Produce BA, and make At ^ AC, and join
CE. . . . .

1. Hyp. Becaufe AE AC, therefore is the
angleACE a z=z E b = f BAC c =r DAC \ dz J# It
therefore DA, CE are parallels, e \Vhe*cfose|) 'u
BA. AE (AC) k BD. DC. c /U

z.Hyp. Becaufe BA. AC (4E):;BD.DC, /d «;f,
therefore are DA, CE parallels j and^ therefore^ 2. 6.
is the angle BAD z& E \ and theangle DAC^ff 2. 6.

i ACE h — E. k therefore the angle BAD r^ g 2a, r
DAC. Wherefore the angle BAC is billed, n 5. r.
Which was to be dem* x.ax.

PROP. IV.

Of equiangular triangles ABC \
DCE, the fides are proportional
which are about the equal angles,
$>DCE, {AB. BC :: DC. CE,
Sec.) And the fides AB,DC,8cc, ,
which are fuhtended under the
E equal angles ACB, £, &c. are

homologous, or of Hi .
Set the fide BC in a direft line to the fide CE,
produce BA and ED till they a meet. a 52. u

H 3 lie-

Digitized by Google

b

c

u8

hyp.
18. 1.

d 34. i.

C 2. 6.

f 16. 5.

Ti* fixth Book of

Becaufe the angle B i = £CDf * therefore
£F, CD are parallel; Alfo becaufe the angle
BOA * — CED, e therefore *re CA, EF parallel.
Therefore the figure CAFD is a Pgr. d therefore
AF r= CD, and AC ~ 4 ED. Whence it is evi*
denr that AB. AF (CD) :: e BCCE./by permu.
taiion therefore AB. BC :: CD. CE. alfo BC.
CL ;: FD (AC.) DE. / and thence by per-
g xx. $. mutation EC AC:: CE. DE. g Wherefore alfo
by equality AB. AC :: CD.DE- Therefore^

Coroll*

Hence AB. DC :: BC. CE :; AC DE.

SchoL

Hence, If in a triangle FBE there be drawn
AC a parallel to one fide FE, the triangle ABC
fhall be like to the whole FBE.

PROP. V.

If two triangles
ABC, DEFbave Heir
fides proportional (AB.
BC:: VE.EF, and AC.
£C::PF9EPy and alfo
AB.AC:: DE.DF) thofc
triangles are equiangu-
lar, and thofc angles e-
fual under whch are fuhtended the homologous Jides. ,

At- tti* fi/1f> W * molro rfiP. ina\* WCl — T '

a a}, r.
b iu i.
cL 6.
dhyp.

At the fide EF a make the angle FEG = Bf
and the angle EFG - C $ * whence the angle
G = A. Therefore GE. EF c :: AB. BC :: DE.
EF. e arid therefore GE DE. Likewiie^GF.

e U.S. and EF e :: AC CB :: d DF. FE. e therefore Qt
9* $• DF. Therefore the triangles DEF, GEF are
f 8. 1. mutually equilateral* / Therefore the angle D=t
% ja. r. G=A, and the angle FED / — FEG = B, and
^font^uentiy *he angle PEF = C There- "

PJtf)P,

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EUCLIDF* Element si St)

PROP. VI.

If t#o triangles ABC.
DEF have one angle 3 ^
equal to one angle uEF,
and the fides about tin
equal angles BJ)EFpro.
/ortional(AB.BC DE.
EF) then thofe triangles
ABC, DEF aie equian-
gular y and havetMe angles equal, under which a) c
■fub tended the homologous fides.

At the fide EF make the angle FEG - B, and
theangleEFG =C$athen willthe angleG-rrA.a jz. i.
Therefore GE.EF :: b AB. BC :: t DE.EF, d andb 4. 6.
therefore DE-GL. But the angle DEF «rrB/~ c hyp.
GEF} therefore the angle D^— G — A, iandd 9. $.
confeQuently the angle EFD - C. W.W.tobcdem. e hyp.
* PROP. VII. te&np.

2| If two triangles ABC, g 4. 1.

DEF have one angle A Crh 31. !•

/ \ f \ the fides about the other
/ / \ angles ABC, £, proportional
fasl > 4f hAB.BC::DE. EF) and

I V JT .By jjij ^ jj

viaining angles C, F «/i*r /f/x or *tf taf* than a
right angle 5 /to; fballtlc triangles ABC, DtFbe
equiangular, and have thofe angles equal about which
the pyoportionaljides are.

For, if ir can be, let the angle ABC cr E, and
make the angle A BG - E. Therefore, whereas
the angle A a — D, b thence is the angle AGB a hyp .
- F. Therefore AB. BG c :: DE. EF :: d AB. b Jl. I.
BC. e therefore BG ' BC. / therefore the angle c 4. 6.
BGC = BCG. g Therefore BGC or C is leisd hyp.
than a right angie, and h confequently AGB or e 9. 5.
F is grater than a right : Therefore the angles f 5. r.
Cand F are not of the fame fpecies or kind, gcor.17.1.
which is againft the Hypothecs. hwavu

^ ' H4 PROP.

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12*

zfyp.
b ri. ax.

d zi. 6.
c i. def.Z.

■

Tie fixtb Book of .
PROP. VIII. .

» *

*

•

If in a right-lined triangle
ABC% from the right angle
BAC there he drawn AD a
perpendicular to the I a ft
^ BC$ then the triangles about
mm T^the perpendicular (JDBy

JDC) are like both to the whole triangle ABCf
and alfo one to the other.

For becaufe BAC, ADB are a right angles, h
and fo equal, and B common ; the triangles
BAC, ADB f are like. By the fame argument
BAC, ADC are ljke d whence alfo ADB, ADC
will be like. Winch was to he dew. ,

Corott.

Hence, i. BD. DA e;-. DA. DC.

z. BC. AC :: AC.DC. and C£. BA ::EA.BD.

PROP. IX..

a jr- *•

c z. 6.
d z8.>

From right line
given AS 4* cut off
any pnrfiequired. as
i (AG.)

• From the point A
draw an infinite line
A C aqy-wife , in
which, a take any
three equal parts
-AD, DE, EF. join FB. to which frbmD idraw
the parallel DG. And the ihinr is done.

For GB. AG c FD. AD ; whence by 4
compofition AB. AG AF. AD. therefore,
whereas AD = -J of AF, therefore is AG = ;
of AB. Which was to be done.

* PROP.

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EUCLIDE'j Elements.

• •

H.I

. PROP. X.

Jo divide ft right line given
AB not divided (in F and G)
as another line given AC rvat
cut (in D and K)

Let a right line RC join
3T G B the extremities of the line
divided, ;md of the line not
divided ; and to that 1 ine from the points E, D
a draw the parallels EG, DF meeting with the a I1- *• ,
right line that is to be cut, in G and F } then
the thing is done.

For let DH be a drawn parallel to AB. Then
AB. DE b AF. FG. and DE. EC b :: DI.J*!*
IH c FG. GB. Winch was to be done, c M-x*«

7. S.

//t-wfr jj learnt to cut a right line given JB into
as many equal farts as you fleafe (fuffofe 5 ;)
which will be more eafily performed tmi*.

Draw an infinite line AD,and another BH pa-
rallel to it, and infinite alfo. Of thofe take equal
parts, AR, RS, S V, VN \ and BZ, ZX, XT, TL 5

ID

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1 22 The fixtb Beak of

in each line lefs parts by one, than are required
in AB ; then let the right lines LR,TS,X\\ZN
be drawn \ thefe lines fo dnwn (hall cut the
right line ^iven AB into five parts,
a jt. t. For RL, ST, VX, NZ are a parallels ; there-
b conjlr. fore, whereas AR, RS, SV, VN are ) e^uair; c
<C Z.6. thence AM, MO, O. , PQare equal alfo. Like-
• wile, ixcaufe that BZ ZX, tueiefore is BQ
*si PQ. and therefoie AB is cut into live paits.
Winch was to be dm.

^ « • iiK I >■' *

PROP. XI.

i

Two right lives being
given JB, JDy to find
out a third in frofouicn
to them (DE.)

foin JtfD, and from
AB being produced take BC — AD. Thro' C
draw CE parallel to DB • with which let AD
prodviced meet in £. then is DE the proportio-
nal required.

^ x. 6. For AR BC (AD) a ;• AU DE. Winch was
to be denu

C Or thus: Make the angle ABC

j/X\ "gnt, and alfo the angle ACD
* A 1 P /Pi****^*, BC.vBC.BD.

or,

PROP.

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£UCLIDFi Elements.

PROP. XII.

Three right linet heing given DE, EFt DG t$
find out a fourth proportional Glf.
r Join EG, and thro' F draw FH parallel to
EG with which let DG produced to H meet.
Then it is evident that DE. fF a :: DG. GH. a x. 6.
Which was to he done.

PROP. XIII.

Two right lines leing given
JE, EE , to find out a mean
proportional EF.

Upon the whole line

E B AB as a diameter defcribe
a femicircle AFB, and f iom E ereft a perpendi-
cular EF meeting with the periphery in F. theft
AE. EF EF. FB. For let AF and FB be
drawn ^ a then from the right angle of the a jt.
right-angled triangle AFB is drawn a right line
FE perpendicular to the bafe. h Therefore AE. b cor.
FE FE. EB. Which was to he done.

i

r.

8.1

CorolL

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124 The fixth Book of

Cor oil.

Hence, A right line drawn in a circle from
any point of the diameter perpendicularly, and
extended to the circumference, is a mean pro-
portional betwixt the two fegments of the dia-
meter.

>

PROP. XIV.

Equal Parallelograms ha-
vi?ig one avgle ABC equal to
one EBG, ha.c the fides BD,
BF which a\ e ah out the equal
angles reciprocal (AB.BG ;;
. [j* EB% BC^ ) And thofe Paral-

lelograms BD.BF which hav$'
;t . one angle ABC equal to one EBG, and the fides-
which are about the equal angles reciprocal, are
• equal.

m rrh t ,For let the fides AB> BG ab<>ut the equal an-
w-gles make one right line ; a wherefore EB, BC

rn < wall do the fame, Let FG, DC be produced
c /• >• till they meet.

e«-S- d BE. BC. e therefore, fife,
I tr ED. BH :: / AB. BG * BE. BC £ .

g Jtf- BF. BH fc Therefore the Pgr. BD^BF. Which

ni.e. wis to he devi. ?

>4 • \ t ^

9. J-

»• • * At

V*

J ■ ♦»

PROP

• i

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EUCLIDE'f Element!.

PROP. XV.

Equal triangles having

one angle ABC equal to one

DBEy their fides which are

about the equal angles arc

reciprocal (AB. BE :: DB.

BC. ) And thofe triangles

that have one angle ABC

equal to one DBEy and have alf$ the fides that arc

about the equal angles reciprocal (AB. BE ;: DB.

BC) arc equal.

Let the fides CB, BC, which are about the - .

equal angles be let in a ftrait line ; a therefore a-/ ,I*,I'f

ABE is a right line. Let CE be drawn. ,

r. tiyp. Ad.BE :; b the triangle ABC.CBE c :: b r' 6*
the triangle DBE.C3E.vriDB.BC.^therefore,&c. ^ 7. 5-

r. Hyp. The triangle ABC. CBE :; / AB. d *• *■
BE g DB. BC h k the triangle DBC. CBE. k % 1 V
Therefore the triangle ABC == DBE. Which1 V6'

h i. 6.
k u, avi

9< 5«

was to be dent .

; r.. •<

PROP. XV I.

[fc

• If four right lines be proportional (AB. FG:: EF.
CB) the rectangle AC comprehended under the ex-
tremes At), CM, is equal to the rcclangle EG com*
prehended under the means FG, EF. And if the re8*
angle AC comprehended under the extremes AB, CB
le equal to the re&anglc EG comprehended under the
means FG, EF, then are the four right lines propor-
tional, (AB. FG :: EF. CB.)

OM i Hyp.

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f 26 The fixth Book of

a 12. 4x. i.Hyp. The angles B and F are right, and *
confeqwently equa!,and by Hyp. AB.FG ::EF.CB.

b f 4. 6. b therefore thei re&angle AC = EG.

c Inp. *• Hyp. The reftangle AC e =s EG, and the

4 14.& angle B = F ; d therefore AB. FG ;: EFT CB.
Which was to be dem.

Corott. ;
Hence, it is eafy t6 apply a reftangle given
t ti.6. EG to a right line given AB ; (viz.) t fey ma-
king A B. EF FG.m

PROP. XVII.

If three right lines he proportional (AB. EP:>
EF. CB.) the reft angle AC made under the extremes
JBy CB is equal to the fyuarc EQ made of the mid-
dle EF. And if the rtttangle AC comprehended under
the extremes AB, CB be equal to the Jquare EQ
made of the middle EF, then the three lines are
poportionat, {AB. EF :: EE. CBJ

Take FG - EF. 7' 4 «

a i Pm 1. Hyp. AB. EF a EF (FG.) CB. therefore
k 16 6 tfce Wangle AC * = EG c = EFq.
c zaJeft *• Hyp.ntht rectangle AC d ~ to the fqfuarc
4 * EG s EFq. e therefore AB. EF FG (EF.) BC.
ti&d. Which was to be dem.

* • • •

Cor oil

Let Ax B Cq. therefore A. C C. B.

PRO?.

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EUCLIDE'i Elements'.
PROP. XVIII. -

V7

B c

From a ri^tr line given AB to defcribe a right-lined
figure AGHB like and alike fituate to a right-lined
figure given CEFD.

Refolve the right-lined figure given into tri-
angles j a Make the angle AhH " D, a and a 2}, t.
the angle BAH — DCF, a ami the angle AHQ
=r CFE, b and the anglt HAG - FcE. then
AGHB fhall be the right- lined figure fought.

For the angle B b D, and the angle b A H b amflu
* = DCF. c wherefore* the angle A HB ■= CFD. c ji. t«
b alfo the angle HAG - FCE and the angle
AHG b CFE. c wherefore the angle G = E,
and the whole angle GAB d = ECD, and the d 2. **.
whote angle GHB d-^. EFD. The Polygones
therefore are mutually equiangular. Moreover
becaufe the triangles are equiangular therefore
AB. BH e;; CD. DF ; and AG GH e :: CE. EF. e 4. 6\
Likewife AG. AH :; e CE. CF. and AH. AB f 22. ^
CF. CD. /From whence by equality AG. AB;; g6.de/* 5-
CF. CD. After the fame mannet GH. HE ;; LF.
FD. Thereforethe Polygenes ABHG, CDEF are
like and alike lituate. Which was to be done.

-

XiX.

Like triangles
ABC, DEF are in
duplicate ratio of
their homologous
fides, B:, EF.

a Let there be si XL 6+
made B C, E F ::
EFjJG. and let AG be drawn* Becaufe that AB.

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j28 The pxth Bdok of

Wor.4*6.DE hi: BC. EF*.v EF. BG, and the angle B=E. '
e eonp. d therefore is the triangle ABGnDEF. But the

1 J.tff* trianSle ABC. ABcft e BC. BG, and/]g = |S

fioir/.j. , ^ r ABC . . ABC BC g
g XX. $. twiceitherefoie^thatiss^=EF twice.

FAjVA to be dem. g

CoroU.

Hence, If three right lines (BC, EF, BG) be
. - proportional, then as the firft is to the third* '
lb is a triangle made upon the firft BC, to *
triangle like and alike defcribed upon the fe-
cond EF: or fo is a triangle defcribed upon the
fecond EF to a triangle like and alike defcri-
bed upon the third.

Like Polygones JBCDE, FGHIK are divided into
equal triangles ABC, FGH, and ACD, FHI, and
ADE, FIKi both equal in number, and homologous
to the wholes (JBC FGH :: JBCDE. FGHIK ::
JCD. FHI :: ADE. FIK) And the foiygonts
JBCDE, FGHIK have a double ratio one to., the
other what one homologous fide BC has to the other
homologous fide GH. T
* . i.For the angle B a=G, and AB. BC a :: FG.

W- QH. b therefore the triangles ABC, FGH
0- ©« are

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EUCLIDE'/ Elenienti.

are equiangular* After. the fame manner are the
triangles A£D,.FKI like. Whereas therefore
the angle BCA b GHF, and the angle ADEb 6.64
h = FiK, and the whole angles BCD, GHI,
and the whole angles CDE, HIK are c equal,c hyp.
there remains the angle ACD d ~ FHI, and the d j.tfj*,
angle ADC = FIH. e from whence alio the e ji, 1,
angle ~CA D = HFI. therefore the triangles
ACD, FHI are like. Therefore, &c.
2. Becaufe that the triangles BCA,GHF are like,
jx\ RCA B C
/thereforeisQ— ~ QHtwice. For the fame rea-f 19. 6.

c . CAD CD . . aUr DEA DE

,S HFI = HI 5 7 nCF = TX
twice, now whereas that BC. GH g :: CD. Big e ;,y* ^
:: D& IK. h therefore is the triangle BCA. f 6 J\
GHF :: CAD. HFI :: DEA. 1KF k :: the poly- hw-ZJ.W

gone ABCDE. FGHIK :: ^ twice. k s«

CbroB.

1. Hfence, If there be three right lines propor-
tional, then as the firft is to the third, fo is a
polygone made upon the firft to a polygone
made on the fecond, like and alike defcribed i
or fo is a polygone defcribed upon the fecond
to a polygone made on the third like and alike
defcribed.

By which is found otit a method of enlarging or
dmmijbing any right- lined figure in a ratio given :
As if you would make a pentagone quintuple of
that pentagone whereof CD is the fide, then be-
twixt AB and 5 AB find out a mean propor-
tional "* upon this raife a pentagone like to* 18. 6«
that given, and it lhall be quintuple of the pen-
tagone given.

2. Hence alfo, If the homologous fides of like
figures be knoufo, then will tue proportion of
the figures be evident, viz. by finding out a third
proposal. PROP.

Digitized

150

Tie y?xfj& £oo£ 0/
PROP. XXL

T

Right lined figures JSC, DIE which are like tcf
the fame right- lined figure HFGf are alfo like one to
the other.

a ief.6. For the angle A rtrH drrD 5 and the angle C
d=Gdz=:E; and the angle Ba~Frt=rI. Alfoa
AB.AC :; HF.HG :: DI.DE.& a AC.CB :: HG.GF
:: DE.EI.And AB.BC:: BF.FG :: DUE.Therefore
ABC, DIE aie alike. Which was to be dem. •

PROP. XXII.

ADO

BC BE Ffii H

If four right lines he proportional (AB. CD ;: EF.
CH) the right-lined figures alfo defcrihed uftn them
leing like and in like fort fituate^Jb all be proportional
(ML CDK:: EM. GO.) And if the right-lined fi-
gures defcrihed upon the lines, like and alike fituate, be
proportional (ABI. CDK:: EM. GO.) then the right
lines alfo (ball be proportional (AS. CD :: EF, GH*)
„ ABI AB . EF . EM.

'•^•^K^^^'ca^^GO
h therefore ABI. CDK :: EM. GO.

AB . ABI. EM EF

* 19. 6*.

b hyp.
c zc. 6.

CD twlCe a ~~ CDK

GO^GH

z.Hyp.

twice. Therefgre AB, CD;; Ef , Oil. Winch wot

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EUCLIDE'; Elements. Jjt

Scbol. m V*

Hence is deduced the manner and reafon ofmulti*
plying futd quantities* ex. gr. Let v 5 l<> ke mul-
tiplied into aJ \. I lay that the produ& jwill be

V 15. For by the definition of multiplication k
ought to be, as 1. ^ 3 :: ^ 5. to the -produft.
Therefore by this q. i.q.'V ? ;: <}• v' ?• 4« of the
produft. therefore the fquare of the produft is
15. Wherefore y' 1 5 is the product of V 5 ^w

V f • FJ/f A irar *0 he denu

THEOREM.

If a right UneAB he cut any-wife in D, thcreSmfctJterig*
Angle comprehended under the parts AD, DB is a
mean proportional hetwixt their fquares. Likewife
the rectangle comprehended under the whole AB and
one part AD, or DB, is a mean proportional hetwixt
the Square of the whole AB and the fquare of the
faid part AD, or DB.

Upon the diameter AB defcribe a femicircle;
from D ere&a perpendicular DE meeting with
the periphery in E. join At. BE.

It's evident that AD. DE a :; DE. DB. b there- a cor. 8.6.
fore ADq DEq:: DEq. DBq. c that is, ADq.b iz. 6.
ADB :: ADB. E)Bq. Wljich was to he dem. c 17. 6.

Moreover B A. AE i:d AE.AD.r therefore BAq.d cor.S. 6\
AEq.v AEq.ADq. /that is, BAq. BAD BAD. e 22. 6.
ADq. After the fame manner ABq.ABD:: AED.f 17. 6.
BDq. Which was to he dem.

Or thus : fuppofe Xrrz A -E. It ismanifeft that
Aq. AE :: a A. E .v a AE. Eq. alio Zq. ZA .v jat.it
Zi A ZA.Aq.andZy.ZE a Z.E:;ZE.Eq.

I % PROP,

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■

y?xf£ Book of
PROP. XXIIL

Equiangular parallelograms
AC) CF\ have the ratio one
to the other , which is
compounded of their fides.
/ AC_BC DC\

VCF~~CG~** CFV
Let the fides about the
equal angles C be a fet in a direft line,and let the
zfch.l$. pgn cil be completed. Then is the ratio of

AC , AC CH _ BC DC m- h

CH ^ CF * ~~ CO "** CE r

bzo.defo.Q2 CH

CoroU.

Jndr. Hence, and from 34, 1. it appears, 1. That trian-
Tacq .i$.5.«?fe* which have one angle equal {as C) have a ratio
compounded of the ratio's of the right lines(AC to CB,

* 3 5. 1. J "^p81 1 1 and * confequently auva-

^ rallelogramsy have their

ratio one to the other com-
pounded of the rations of
bafe to bafe% and altitude
to altitude. Aftef the
like manner you may
argue in triangles.

From hence is appa-
rent how to give the proportion of triangles ana pa-
rallelograms. Let there be two Pgrs. X and Z»
tvhofe bafes are AC, CB, and altitudes CL, CF,
♦14. 6. Make CL. CF CB, Q. * theft *ill it beX.Z.v
*ndt<6. ACQ,

L

A C

O

■

*
•
«

■

4

*

•

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*3?

EUCLIDE*/ Elements.

• PROP. XXIV,

In every parallelogram ABCD,
>be parallebgtams EG, HF winch ;
hre about the diameter AC are like
to the whole, and alfo one to the
other.

For the Pgrs. EG, HF have
each of them oneangle common with the whole,
a therefore they are equiangular to the whole, a 2n i
and alfo one to the other. Alfo both the trian-
gles ABC, AEI, IHC a and the triangles ADC, .
AGI, IFC are equiangular mutually ; b therefore b 4* *»
AE. EI AB,BC, and b AE. AI AB. AC, and
h AI. AG AC. AD. c Therefore by equality, * "t *•
AE. AG :: AB. AD. d Therefore the Pgrs. EG, lJcf<f
BD are like. After the fame manner are HF, •
BD like alfo. Therefore, &c.

Unto a right-lined figure given ABEDCto defcribe
another figure P like and alike fituate, which alfo Jball
be equal to another right-lined figure given F.

a Make the re&angle AL — ABEDC ; b alfo a 4 $. U
upon BL make the re&angle BM^F ; betwixt b 44. 1.
AB and BH c find out a mean proportional NO; c 1 3. 6.
Upon NO b make the polygone B like to the d 18. <5f
nght-lined figure given ABEDC. I fay the po-

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Ig4 7&< y?x/i Ztoai 0/

t ro> Mo.6.]ygone P fo made fhall be equal to F that was

g 14 5. Per ABEDC (AL) P.vtf A8.BH::j At&
fc foatfr. BM . Therefore ?g --. BM. * = F, Which was>
to be done.

PROP. XXVI.

If from a Parallelogram ABCD,
ie taken away another Parallelo-
gram AGFE, like unto the wlnle%
and in like fo»t fety having alfo an
angle common with it EAG ; then
is that Parallelogram about the fame
diagonal AC with the whole.
If you deny AC to be the common diagonal,
then let AHC be it, cutting EF in H, and let
HI be drawn parallel to AE.Then arePgrs.EI,DB
I z4- 6- a like, b therefore AE.EH.v AD.DC ::c AE.EF.
P fj^W* and d confequently EHrrEF. f Winch is abfurd.

PROP. XXVIL

c

do. 5.
f 9. ax+

m -

"NT E ®f °^ parallelograms 9
AD, AG applied to the
fame right line ABy and

parallelograms CE% KI like
and alike fet to the Pgr, AD

C J£

D 1. ax.

C 16. U
<J h ax.

1 2. ax*

half line, the greatefi is
that AD which is applied to the half being like to
the dsfcll KL

For becaufe that GE a = GC, and KI added
in common, b thence is KE t=z CI c z=z AM.
add CG in common, d then is AG = to the
Gnomon MBL. But the Gnomon MBL e ~y
CE (AD.) Therefore AG -3 AD. Winch was
to be demonjlrated.

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EUCLIDE'j Elements'*
PROP. XXVIII.

*3f

Upon a light line given AB, to apply a parallels
gram AP equal to a right-lined figure given C, defi-
cient by a parallelogram ZR which is like to another
parallelogram given D. * Now it is requifite thatthe * 2,7. &
night-lined figure given C, whereunto the Pgr. to be
applied AP muft be equals be not greater than the
Pgr. JF which is applied upon the half line, the de-
fers being like, namely the defeil of the Pg>: AF9
which is applied to the half line, and the defed of the
Pgr. D to be applied whofe defect is to be like to the
Pgr. given.

Bifeft AB in E ; upon EB a make the Pgr. a 18.6.
EG like to the Pgr. I) ; and b let EG — C-^ I. bfch.tf.u
c Make the Pgr. NT ~ I, and like to the Pgr. c x$#6.
given D, or EG 5 Draw the diameter FB } Make
FO = KN, and FCl~ KT b thro' O and
draw the parallels SR, QZ. Then is the Pgr.
AP that which was fought.

For the Pgrs. D, £G, OQ., NT, ZR are all d dconj!r.&

like one to the other, and the Pgr. EG — e NT 24* &

-|- CrrsOQ^-C. /wherefore C — to thtcconP-

Gnomon OBQ g -9 AO -+ PG = h AQ + $P f 3- ax*

S AP. Which was t» be done. **V

114$. 1.

PROP.

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■

The fixth Book of
PROP. XXIX.

a 18. 6.
c j. i.

i

dconftr.
c 24. r.
f conjlr.

«*•
n i<
k 45. r.

ax.

Upon a right line given JB. to apply a Pgr. JN
equal to a ngbP lined figure given C, exceeding by a
Pgr. OP, which Jl) ail be like to another Pgr. given D.

Bifeft AB in £. Upon EB a make a Pgr.EG
like to D, which was given. Andi let the Pgr.
HK ~ EG-*C,and like to D given, or EG. Make
FELr-c IH, and c FGM— IK. Thro' L,M draw
the parailelsMN and RN,*nd AR parallel toNM.
produce ABP, GBO. Draw the diameter FBN.
Then is AN the parallelogram required.

Forthepgrs. D,HK.LM.EG are d like, e there-
fore the pgr. OP is like to the pgr. LM, or U.
Alfo LM fsz HK /— EG -\- C. g Therefore C—
to the Gnomon ENG. But AL /;~LB kzzBM.
I therefore C =z AN. Which was to be done.

PROP. XXX.

\

To cut a finite right
line given JB according
H to extreme and mean
J ratio (JB. JG :; JG.
I GB.)

a Cut AB in G,
in fuch wife that
FAB x BG = AGq. J,

Then BA. AG :: AG. GB. Which was to be done.

PROP.

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EUCLIDE'* Elements.

'17

In right-angled triangles BJC, any figure BF de-
ferred upon the fide BC fuhtending the right angle
BACy is equal to the figures BG.ALy defenbei ufpn
Hie fides BJ9 JCj containing the right angle, like
and alike fituate to the former BF.

From the right angle BAC let down a per-
pendicular AD. Becaufe that DC. CA :; a CA.
CB, b therefore AL. BF DC. CB. Alfo, be-
caufe DB. BA a BA. BC, h therefore BG. BF

DB. BC. c therefore AL BG. BF :: DC +■
DB (BC.) BC. d Therefore AL -4- BO = BF.
Which was to be dem.

Or thus : BG. BF ;; e BAq, BCq. And e AL,
BF ACq. BCq. /therefore BG AL. BF.v
BAq -4- ACq. BCq. g Therefore whereas BAq -*-
ACq—h BCq. h thence is BG+AL=BF. Wbicl
was to be dem.

i

CoroU*

» i *

From this Propofition you may learn how to
add or fubftraft any like figures, by the famit
method that is ufed in adding and iubftia&ing
q{ fquares, in Schol. 47. 1.

PROP.

a cor.ZAT
b cor.

f x4. 5.
to 47. J.

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The fixth Book of

PROP. XXXIL

If two triangles ABC,
DCE having two fides
propational to two (AB.
AC DC DE.) he fo
compounded or fet toge-
ther at one angle ACD
that their homologous fides
le alfo parallel AB to DC, and AC to DE,) then the
remaining fides of thofe triangles (BC, CE) Jball b&
found placed in one fir ait line.

For the angle A a = ACD a — D, and AB.
AC b :: DC. DE. c therefore the angle B=DCEf
Therefore the angle B | A d = ACE. But the

anS 6 B.^AC^ e = * right- /"therefore the
angle ACE + ACB — z right, g therefore BCE
*> a right line. Which was to btdem.

PROP. XXXIII.

In equal circles DBCA, HFGPy the angles BDC,
FHG have the fame ratio with their peripheries BC9
FG on which they infifi \ whether the angles be fet
at th* centers {as BDC, FHG) or at the circumfe-
rences, Ay E : And in like fort are the feSonBpC$
FHG, beemfe defcribed vp<in> th* cmtm>

i •

Dr*w

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EUGLIDEV Elements. \

Draw the right iines BC, FG. ^i—CB^
an4 GL—FG - LP, and 'y>'<« HL, HP.

The arch BC a~Cl. a alfo the arch FG,GL, a x& gm
LP are equal, £ therefore the angle BDCrrCDI, b 27. J.
* and the angle FGH^GHL- LHP. Therefore
the arch BI is as multiple of the archBC,asthe
angle BDI isof the angle BDC. And in like man-
ner is the arch EP as multiple of- the arch FG,
as the angle FHP is of the angle FH<3. But if
the arch BI Er,r=:,~3FP. c then likewife is the
angle BDI cr, FHP. Therefore is the : wch d ^ f , ,

BC.FG;; d the angle BDC.FHG ' V~?15' $m '

/:; A. E. 7^ic7; w/« *o Aw-

Moreover, the angle BMC^ — CNI ; iand g 27. 3.
therefore the fegment BCM ~ CIN. k Alfo the h 24. J.
triangle BDC — GDI, I wherefore the feftor k 4. 1.
BDCM = CD1N. Alter the lame manner are 1 z.<tx.
the fectors FHG, GHL, LHP equal one to the '
other. Therefore fince accordingly as the arch
BI tr, rr, "3 FGP, fo is likewife the feftor
BDI cr, =r,~3FHP , m thence fliall be the fedor m
BDC. FHG :: the arch BC. FG. Which was to
be demonftrated.

Corolf.

1. Hence, As f eft or is to fed or f fo is angle to %u y#

z. Tfo angle BBC m the center is to four right
angles, as the arch BC, on which it injifts, to the
whole circumference.

For as the angle BDC is to a right angle, fo
is the arch BC to a quadrant. Therefore BDC is
lo to four right angles as the arch BC is to four
quadrants, that is,the whole circumference. Alfo
the angle A. 2 right :: the arch BC periphery, ' . r'

3. Hence, The arches IL, BC of unequal wcUt.
which fubtend equal angles, whether at the centers')
* IJL and BAC, or at the Periphery m like.

f f For

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The fixth Book of

♦ For II. p^rioh. :: angle IAL (BAC) 4 rights
Alfo Arch BC. periFi» :; angle BAC, 4 right.
Therefore IL. periph::'BC. peri ph. And confe-
quently the arches IL and BC are like. Whence

4. Twofmidiamcters ABy AC cut off like arches
IL> BCfrom concentrical peripherics.

■ >

The End of the fixth Book.

THE

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THE SEVENTH BOOK

OF

E UCLI.DE/ ELEMENTS.

Definitions.

1. m *m Nity is that, by which every thing

that is, is called One.

II. Number is a multitude com-
pofed of units.

III. One number is a part of
another, the leffer of the greater, when the lef-
fer meafures the greater.

Every fart is denominated from that number by
which it meafures the number whereof it is a fart \
as 4 is called the third fart of 12, becaufe it mea-
fures 1 1 hy j.

IV. But the leffer number is termed Parts,
when it meafures not the greater.

All farts whatfoever are denominated from thofe
two numbers, by which the great eft common meafure of
the two numbers meafures each of themyviois faid to be
\ of the number 15} becaufe the greateft common mea-
fure ^ which is $ j meafures 10 by z, and 15 by 3.

V. A number is Multiple (or Manifold) a
greater in comparifon of a leffer, when the leffer
meatures the greater,

VI. An even number is that which may be
divided into two equal parts.

VII. But an odd number is that which can-
not be divided into two equal parts j or, that
which differs from an even number by an unit.

VIII. A number evenly even is that which
an even number measures by an even number.

IX. But a number evenly odd is that which
an even number meafures by an odd number.

X.A

The feventh Book of

X. A number oddly odd is that which an odd
number meafures by an odd number.

XI. A prime (or firft) number is that, which
is meafured only by an unit.

XII. Numbers prime the one to the other,
are fuch as only an unit doth ineafure, being
their common ineafure.

XIII. A compofed number is that which fome
certain number meafures.

XIV. Numbers compofed the one to the other,
are they, which fome number, being a common
ineafure to them both, does mcafure.

In this, and the preceding definition, unity is net
a number.

XV. One number is faid to multiply another,
when the number multiplied is fo often added
to it felf, as there are units in the number mul-
tiplying, and another number is produced.

Hence in every multiplication a unit is to tlx mul-
tiplier, as the multiplicand is to the produtt.

Obf. That many times, when any number are to

letters denotes the-produtt ; So AB ^ Ax B, and
CDE =± C x D x E.

XVI. When two numbers multiplying them-
felves produce another, the number produced is
called a plain number \ and the numbers which
multiplied one another, are called the fides of
that number : 6a i (C) x 3 (D) = 6= CD is a
pain number.

XVII. But when three numbers multiplying
one another produce any number, the number
produced is termed a folid number ; and the
numbers multiplying one another, are the fides
thereof : So z (C) x j (D) x 5 (£) = 30 = CDE
is a folid number.

XVIII. A l^uare number is that which is
equally equal ; or, which is contained under
two equal numbers. Let A be the fide of a fquare ;
th$ fquare is thus noted, AAy or Ag. or J*.

XIX. A

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EtJCLIDE'i Elements.

XIX. A Cube is that number which is equally
equal equally, or,which is contained under three
equal numbers.!** A be tlx fide of a CuhrjU Cube
is thus noted, AAA, or Ac. or A '> /

In this definition, and the thee foregoing, unity
is a number.

XX. Numbers are proportional, when the firft
is as multiple of the fecond, as the third is of
the fourth j or, the fame part; or, whmapart
of the firft Dumber mealurti the fecond, and the
fame part of the third meafures the fourth, e-
qually : and on the eoiuia.y. fio A. B C D9
that is, j. o :: 5. 1 5.

XXJ. Like plane, and folid numbers are they,
which have their tides proportional : Namely]
not all the fides, but fome.

XXII. A perfect number is that which is equal
to all its aliquot parts.

As 6 and 28. But a number that is lefs than its
aliquot-parts is called an Abounding number 5 and a
greater a Diminutive : Jo 12 is an Abounding, 15
a Diminutive number.

XXIII. One number is faid to nieafure ano-
ther, by that number, which, when ir multi-
plies, or is multiplied by it, it produces.

In Divifion, a unit is to the quotient as the d\vi-

for is to the dividend. Note, that a number placed

under another with a line betwixt them,figmfies Du

A - *i 4 CA
Won; So g= A divided by E, and-g-— G x A

divided by B«

Two numbers are called Terms or Roots of
Proportion, lelTer than which cannot be found
in the fame proportion.

Tcftulates or "Petitions.
Hat numbers equal or manifold to any

number may be taken at pleafure.
at a greater number may be taken than
number whatfoever.

*• That

w

Tie feventh Book of

j. That Addition, Subftraftion, Multista-
tion, Divifioj^ and the Extra&ionsi of roots or
fides of fquire and cube numbers, be alfo grant-
ed as-pomble,

AxiomeL

, _ agrees with one or msiny equal

w w numbers, agrees likewife with the reft.
2. Thofe parts that are the fame to the fame
part, or parts, are the fame amongft themfelves.

2. Numbers that are the fame parts of equal
numbers, or of the fame number, are equal
amongft themfelves.

4. Thofe numbers, of whom the fame num-
ber, or equal numbers, are the fame parts, are
equal amongft themfelves.

5. Unity ineafures every number by the units
that are in it \ that is, by the fame number.

6. Every number meafures it felf by a unity.

7. If one number, multiplying another, pro-
duce a third, the multiplier lhall meafure the
product by the multiplicand ^ and the multipli-
cand fliall meafure the fame by the multiplier.

Hence, No prime number is either a plane , folid,
fauarty or cube number.

8. If one number meafures another, that number
fcy which it meafures fhall meafure the fame by
the units that are in the number meafuring,
that is by the number itfelf that meafures.

9. If a number meafuring another, multiply
that by which it meafures, or be multiplied by itf
it produces that number which it mealurcs.

10. How many numbers foever any number
meafures, it likewife meafures the number com-

pofed of them.

11. If a number meafure any number, it alio
meafures eveiy number which the faid number
iiiCciiurcs,

iz. A number that meafures the whole and a
part taken away doth alfo meafure the icfiduc.

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EUCLIDE'* Elenunts. jtf

PROP. L

A E..G.B 8 s i If two unequal num-

C...F..D i I i hers AB , CD, being given,
H--- the lejfer CD be conti-

nually taken from the
greater AB (and the refidue EB from CD, &c.) by
an alternate fubjtraftion, and the number remaining
do never meafure the precedent, tiU the unity GB be
taken ; then are the numbers which we} e given AB,
CD, prime the one to the other.

If you deny it, let AB, CD have a common
meafure, namely, the number H. Therefore H
meafuring CD, does a alfo meafure AE ; and* tt.dx.j.
confequently the remainder EB ; a therefore it
Jikewife meafures CF, and b f o the remainder b I2.0x.74
FD ; a wherefore it alio ineafures EG. But it
meafured the whole EB, and b therefore it mull
meafure that which remains GB, a unity, itfelf
being a number, c Which is abfura\ c p.ax* f4

P R O P. II.
9 6 Two numbers A B,

A ....E B 15 9 6 CD being given, not

6 3 prime the one to tie

C F...D # r r other to jind out tlxir

G — great eft common mea-

fure FD.

Take the lefler number CD from the greater
AB as often as you can. If nothing remains, a% G.dx.ji
it is manifeft that CD is the greateft common
meafure. But if there remains fomething (as EB)
then take it out of CD, and the refidue FD out
ofEB, and to forward till fome number (FD),
meafure the faid EB (b for this will be before0 *• /•
you come to a unity.) FD fhall be the greateft
common meafure. a

For FD <; meafures EB, andi therefore alfo C£ ; • eoVr' .
and e confequently the whole CDyd therefore like- ■ "•**»7*
wifeAEjand fomeafures the whole AB. Wherefore* 12«**-7*
. i i J * i K it

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1^6 Tie feventb Book of

it is evident that FD is a common meafure. If
you deny it to be the greateft, let there be a
d greater (GO then whereas G rtaeafures CD, it d

tiutxsj. muft Jikewife meafure AE,tf and the refidue EB,

Sfuppof. d as alfo CF, e and by confequence the refidue
9. ax. 1. FD, g the greater the lefs. h Which is abfwd.

CoroU.

Hence, A numcer that meafures two numbers,
does alfo meafure their greateft common meafure.

PROP. III.

JZ Three numbers being given, J,B,

B.~~.8 C, not prime to one another to find

D. ...4 out their greateft common meafure
\Qt 6 £•

E. .i Find out D the greateft com-
F — mon meafure of the two numbers

A, B. If D meafures C the third,
it is clear that D is the greateft common mea-
fure of all the three numbers. If D does not mea-
fure C, at leaft D and C will be compofed the
one to the other, by the CoroU of the Prop, pre-
ceding. Therefore let E be the greateft common
meafure of the faid numbers D and C, and it
appears to be the number reouired.

a eonjtn *or E a meafurcs c and p3 and ^ meafures
b luaxi. A and B ^ therefore b E meafures each of the
*' numbers A, B,C; neither fliallany greater (F)
.1. 7. rneafure them 5 for if you affirm that, c then
* F meafuring A and B, does likewife meafure D
their greateft common meafure ; and in like
manner, F meafuring D and C, does alfo mea-
d fuppof-l fure E c their greateft common meafure, d the
e 9. ax* 1. greater the lefs. e Which is abfurd.

[CoroU.

Hence, A number that meafures three Rum*
bers, does alio meafure their greateft cpromon
wieafure.

1 1

tear.

PROP.

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EUCLIDE'j Elements, 147

PROP. IV,

A .. y.. 6 £ rcry fcr/> number A is of every

B 7 greater B eitljer a part or parts.

B 18 If A and B be prime to one

B 9 another, a A flull be as many a ±itfj*.

parts or the number B, as there
are unities in A (as 6 = * 7.) But if A mea-
iures B, it is b plain that A is a part of B (as 6 b i*def.j*
= J 18.) Laitly, if A and B be otherwife com-
pofed to one another, c the greateft common q ^def-f*
meafure fhall determine how many parts A does
contain of B j as 6 ■=> \ 9.

PROP. V.

A 6 D .... 4

6 6 4 4

K ...... Cj ...... C 1 z, E .... H .... t 8.

If a number A be a part of a number BC. and
Another number D the fame part of another number
EF } then both the numbers together (A + D) JbalL
he the fame part of both the numbers together ( BC
•+EFJ which one number A is of one number BC.

For if BC be refolved into its parts BG, GC*
equal to A ; and EF alfo into its parts EH, HF
equal to D $ a the number of parts in BC fhall a hyp.
be equal to trje number of parts in EF. There-
fore finceA+D^BG+EH-GC+Htf, thence bccnfi.ani
A-+D fhall be as often in BC -J- EF, as A is in ax* t.
BC* Which was to be dem.

Or thus. Let a = — , and br=-^ theni.arzx, c %i ax, |<

and 2 b=y* wherefore **-^tb^zx+y* there*.

fore a ^ b — - JPfoVi mi to be Am.

x

K *

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148 The feventh Book of

PROP. VI.

I j 4 4 V a number AB

A...G...B6 D....H....E8 be parts of a mtm-

C .9 F iz her C9 and another

number DE the fame
f arts of another number F^ then both numbers together
JB-4-DE Jbali be of both numbers together C -+F the
fame pat ts, that one number AB is of one number C,

Divide AB into its parts AG, GB; and DE in-
to its parts DH, HE. The multitude of parts in
*• both AB, DE is equal by fuppofition ^ wherefore
a fince AG a is the fame part of the number C,

that DH is of the number F ; therefore AG -\-
b $• 7- DH b ftiall be the fame part of the compounded
number C-»-F, that one number AG is of one
number C, b In like manner GB-*-HEis the fame
part of the faid C -J- F, that one number GB is
t %.ax* 7. of one number C. c Therefore AB -| DE is the
fame parts of C -+ F, that AB is of C. Winch
was to be dem.

Or thur. Let a ss jf x, and b = * y, and x-+-
y '= g, then, becaufe 5 a^2x, and 3 b — 2 y,
is$a-+-3b"z,x-+-z,yr=: zg. therefore a-h

b = i g = !* x-y-

PROP. VII.
$ j If a number AB be

A E...B8 the fame part of a

6 10 6 number CD, that a

G •••••• 0 HMWHH F •••••* D16 part taken away AE

is of a part taken away
CF\ then fi; all the refidue EB be the fame part of the
tin refidue FD,that the whole AB is of the whole CD*
a t.poft.j. a LetEB be the fame part of the numberGCthat
b %. 7. AB is of CD.or AE of CF. b Therefore AE+EB
is the fame part of CF +GC that AEis otCF,or
c 6. ax, 1. AB of CD. c therefore GF = CD. take away
d 3. ax a, CF common to both, and d there remains GCrr
e 1. ax.0], FD. e "Wherefore EB is the fame part of the re-
fidue FD (GC) that the whole AB is of the
whole C& Which was to be dm.

Or

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EUCLIDE** Elements. tiff

Or thus. Let a -+ brrx- and c-»-ic=y ; and
y, in like manner as a~ } c ; I lay b~ $ d.
For j c-+i d/=:jy'x^r=a^b. takeaway from f i. 2.
both * eg = a, and £ there remains j d = b.g byf.
Which was to he devu

PROP. VIII.

6 2 4 2 2. /jf # numher AB

A......H..G.... E..L..B 16 J* //;e /awe jwm of

18 6 j number CD, /Ad/ a

C.~........~. ...F D24 part taken away JE t.

is of a fart taken
away CF ; the rejidue alfo IiB JbaU he of the re-
fidue FD the fame parts, that the whole AB is of
the whole CD.

Divide AB into AG, GB, parts of the number
CD ; aHb AE into*AH, HE, parts of the num-
ber CF; and take GL~ AH—HE. a wherefore a j. a*.*.
HG^EL. And becaule h AG-GB, c therefore* conflr.
HG—LB. Now whereas the whole AG is the^ 3.*»«i.
fame part of the whole CD that the part taken
away AH is of the Dart taken away CF, d the « 7« !•
refidue HG or EL mail be the fame part alfo
of the refidue FD that AG is of CD. In like
manner, becaule GB is the fame part of the
whole CD, that HE or GL are of CF, d there-
fore the refidue LB {hall be the fame part of the
refidue FD that GB is of the whole CD. There-
fore EL -+ LB (EB) is the fame parts of the re-
fidue FD, that the whole A B is of the whole .
CD. Winch was to he dem. ' '

Or thus more eafily. Let a V b~x,andc-+d:=y.
Alfo y=s| x as well as c = -* a } or, e which is -
the fame, 3 yr=2 x ; and 3 c r= 2 a. I fay d = e ax%'9
i b. For $ c -h 3 d /= 3 y = 2 x/= 2 a + if X lm

1*

PROP.1*"*' *

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J je Tie Jtvtnth Book of

PROP. IX.
.A*.., 4 If a number A he a -part of a.
», 4 4 number BC, and another number

B. ..G....C8 D the fame part of another number
5 D..... $ BF\ then alternately what part or
5 S farts thefirftAis of the third D, the

E H F io /aw* fart or parts JhalL the fecond

• £C iff /our** 27F.

i A is fuppofed "3 D. therefore let BG,GC,and

EH,HF,parts of the numbers BC,EF be equal; BG

and GC to A; and EH, HF to D. Themultitude

of parts is put equal in both. But it is clear that

a i.ax. 7- GB is a the fame part or partsof EH, that GC is of

and 4. 7. HF; iwherefoie BC (BG-t-GC) is the fame part

P }'&6'7*oi pans of EF (EH-+HF) that BG alone (A) is

of EH alone (D.) Which was to be dem.

b d*
Or thus. Let a— and c = — ? or ? a sc b and

iCrrd. then is- = l? = i
a a ga b#

PROP. X.

A G .. B 4 If a number AB be parts of a

C 6 number C, jwi another number

5 5 *#ff fame parts of another

D. ... H E 10 number t\ then alternately, what

F i$ part or parts the Ji>Ji AB is of

the third DE, the fame parts or

* part Jball the fecond C be of the fourth F.

AB is taken -3 DE, and C^F. Let AG,GB,

and PH, HE be parts of the numbers C and F,

viz. as many in AB,asin JOE. It is manifeft that

ft 9. 7- ' AG is the fame part of C, that DH is of F. a

whence alternately AG is ofDH,and like wife GB

P5.&9-7-of HE, and b fo conjointly AB of DE the fame

part,orparts,that Cis of F. Which was to be dem.

^1 , 2b 2d

Or thus, Leta= ~,&c= — • or 5 a 2 b, &

3 35
, . c :c 2d d

30= id, Then is — = ~ — t-

a 3a 2b b.

PROf,

4

1

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EUCLIDE'i Elements. ifi
PROP. XI.

If a part taken away AE
47 le to a fart taken away CF9

A E B 7 as the whole AB is to the

8 6 whole CD, the refidue alfo

C. ....... F D14 EB /ball be to the refidue Fl),

as the whole A* " to the
whole CD. a a. 7.

Firft, let AB be ~3 CD ; a then AB is either b zojefr.
a part or parts of the number CD ; and hkewile c 7>or<g>7>
A£ is* the fame part or parts of CF ; c therer.
the refidue EB is the fame part of parts of the
refidue FD, that the whole AB is of the whole
CD. h and fo AB. CD :: EB. FD. But if AB be
trCD, then according to what is already lhewn,
will CD. AB :: FD. EB. therefore by inverfion
AB. CD :: EB. FD.

PROP. XII.

A, 4. C,2. E, 3. If there be divers numbers,

B, 8. D, 4, F,6. how many foever, proportional

(A.B::C.D::E.F,)tben

as one of the antecedents A is to one of the confer
quents B, fo Jball all the antecedents (A-+C-\-E) be
to all the confequents (B-+D-+F.)

Firft, let A, C, E, be B, D, F i then (be- ^ _
caufe of the fame proportions) a fhall A be the JJL
fame part or parts of B that C is of b and u
likewife conjointly A-+C lhall be the fame part
or parts of B+D that A alone is of B alone. In
like manner A-+C+E is the fame part or parts of 7
B+D+Fthat A is of B. c Therefore A+C+E.c M ' '
B+D-+-F :: A. B. But if A, C, E,be put greater
than B, D, F, the fame may be ftiewn by in-
verfion,

K4 **0P-

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I $* Tie feventb Book of ' I

PROP. XIII.

A, J. C, 4. If there be four numbers proportional

B, 5. D, 12. (J.B::C. D.) then alternately they

Jballalfo be proportional, (A.C::BJ).)
Firft let A and C be^B and D, and A -3
a zoMtf. c. By reafon of the fame proportion a flull A
P9-CT 10. be the fame part or parts of B, that C is of D.
7- .7.1 h Therefore alternately A is the fame part or
parts of C that B is of D. and fo AX :: B. D,
si , But if A becr.C, and A and C fuppofed -3 B
and D, the matter will be the fame by invert-
ing the proportions.

PROP. XIV.

A, 9. D, 6. If there be numb ersjjow many foever,

B, 6. £,4. A, B, C, and as many more equal to

C, F, 2. i?z multitude, which may be com-

pared two and two in the fame propor^
tion(A.B::D.E.andB.C::E.Fh) they Jballalfo, of
equality, be in the fame -proportion (AX:: D. F.)
113.7. For becaufe A. B :: D. E. a therefore alternately
is A. D:: B. E::tf C.F. a and fo again by chan~
ging A. C :: D. F. Which wa* to be dem.

PROP. XV.

' Jj If a unit mea/ure any number B9

B».» J -E 6. and another number D do equally

meafurefome other number E ; al-
ternately alfojball a unit meafure the third number
D9 as often as the fecond B docs the fourth E.

0 7 *For fedng 1 is the fame Part ot B> that D *
/• of E j a therefore alternately fhall 1 be the

fame part of D, that B is of E, Wbkb ms to be
iemonjlrated.

* PROP.

« •

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EUCLIDE'* Elements,

PROP. XVI.
B, 4. A, t. If two numbers, A* B, mnlt\-

A, \. B, 4. plying therkfekei the one into
AB, II. BA,iz. the other, produce fifty numbers

AB, BA\ the muni en produced
AB and BA Jball be equal the one to the other.

For becaufe AB rz A x B, a therefore ihall 1 a * J-*/-7-
be as often in A, as B in AB, b and by confe-^ IS«7- *
quence alternately 1 fhall be as often in B as A
in AB. But for that BA ^ BxA, a therefore
ihall 1 be as often in B, as A in BA. therefore
as often as 1 is in AB, fo often is r in BA. and
c fo AB = BA. Which was to be dem. c 4. ax. 7.

PROP. XVII.
A,}. If a number A multiplying

B, z. C, 4. two numbers By C, produce other

AB, 6. AC, il. numbers, AB, AC ; the numbers

produced of them JbaU be in the •
fame proportion that the numbers multiplied are* r
(AB.AC::B.C.)

For being AB=^A xB, a therefore fhall 1 be as a 1 $.def*j
often in A, as B in AB. a Likewife becaule AC
= AxC, therefore ihall 1 he as often in A, asC
in AC. and fo alfo B as often in AB as C in AC.
i wherefore B. AB :: C.AC, c and therefore alfo b ze.def.j
alternately B.C.*: AB. AC. Which was tobe dem. c 13. 7*

PROP. XVIII.

C, 5. C, j. // two numbers A, B, mul-
A, B, 9. tiplying any number C\ produce

AC, 15. BC~45. other numbers AC, BC ; the

numbers produced of them JbaU
be inthe fame proportion that the numbers multiply-
ing are. (A. B ;: AC. BC)

For*AC~CA, andECa^CBj fo the fame a 16. 7-
C multiplying A and B produces AC and BC.
I therefore A. B AC.BC. WTiUb was to be dem. b 17. 7.

Scbol.

1

„ Digitized

ij4 73« /SvortF AwE */

Schol,

Hence is deduced the vulgar manner of redu-
cing fraftions (|, to the Ume denomination.
Fox multiply 9 both by 5 and j, and they pro-
duce£f:=^ becaufeby this, 5. 5 :: 27.45. Like-
wife multiply $ by 7 and 9^ there arifes * j -3
2; becaufe7,9:: 35*45-

PROP. XIX.
A , 4. B, 6. C, 8. D, 1 z. If there be four num-
AP,4& BC, 48. hers in -Proportion (J.

B ;: C. JD) the number
podueed qftbefirjt and fourth (JD) is equal to the
number which is produced of the fecond and third
(JBC.) And if the number which u produced of the
firft and fourth (JD)be equal to that produced of the
fecond and third (BC) thofe four numbers Jbat be in
proportion CJ.B ;: CD.)
a 17. 7. t.ffyf. ForAC.ADa:: C.Db:: A.Bcr. ACBC.
b {J * therefore AD=BC. Which was to be dem.
c i& 7. *• tyh Becaufe e AD—BC, therefore AC. AD
d 9. 5. /:: AC~ BC- But AC.AD^ :: C. D. and ACBC
c hyp. * * A* B. k therefore C. D :: A.B. Which was to
f y# j# 4* demonjlrated.

hi8.7. PROP. XX.

k <* ^# ^# If there be three numbers in

4. 6. 9. proportion (J. B :: B. C) the
AC, 36. BB, 56. number contained under the ex-
D, 6. /www (JC) is equal to the
fquare made of the middle (BB,)
Jnd if the number contained under the extremes be
equal to that (Bq.) produced of the middle, thofe thoe

numbers fiaU be in proportion

a uax.7. 1. Hyp. For take D=B. a therefore A.B :: D
b 19. 7* (B.) C. b wherefore AC=BD, a ot BB. Which
wm to be demonjlrated.

z. Hyp.

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EUCLIDE'* Elements. Jjy

2. Hyp. Becaufe AC c = BD, d therefore A. c byp.
B:: D (B) C. Which was to be dent. d 19. J.

PROP, XXI.

A...G..B5, E 10 Numbers ABy CD,

C „ H.D3. F 6 the leafi of all

that have the fame
proportion with them (£, F) do equally meafure the
numbers, Ey F, having the fame proportion with
them • the greater Ad the greater Ey and the lejfer
CD the lejjer F. ,

For AB. CD a :: E. F. b therefore alternately ,
AB. E :: CD. F. c therefore AB is the lame part ? ^
or parts off. that CD is of F. Not parts; for if **Zlm
fo, then let AG, GB be parts of the number E 5 C Z0MJ^
and CH, HD, parts of the number F. e therefore
AG. E :: CH. F. and by inverfion AG. CH d :: d l*. 7.
E. F e :: AB. CD. therefore AB, CD are not the « tyt*
leall in their proportion ; which is contrary to
the Hypothefis. Therefore, g>V.

PROP. XXII.

A, 4. D, 12. If there be thee numbers A,
By 3. E, 8. B9 C ; and other numbers equal 1
C, 2. F, 6. to them in multitude, D, Ey Ff

xriif 4 wj/ compared two and
two in the fame proportion : And if alfo the pro-
portion of them be perturbed {A. B :: E. F. and B.
C :: D.E.) then of equality they Jballbeinthe fame
proportion {A. C :: D. F.) ' 7

For becaufe A. B a :: E. F, therefore (hall A F *
■5= BE ; and becaufe B. C :: a D. E, b theiefore b '*
BEmCD. c and confequetatly A Fm CD. d W here- ci.ax. t%
fore A. C D. F, Winch was to be dem. 0*9.7,

PROP.

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If6 ' The feventh Book of

■ < PROP- XXIII. h

A, 9. B, 4. Numbers prime the one to the

C D — other , J, B, are the leaft of all

E-- ?mmbers that have the fame pro-
portion with them.
If it be poflible, let C and D be lefs than A
a 21.7. and B, and in the lame proportion ^ a therefore
C meafures A equally as D meafures B, namely,
b i^Jefjiby the fame number F ; and fo C {hall be h as
c 15. 7. often in A as 1 is in E \ c like wife alternately,E
as often in A as 1 in C. By the like inference as
many times as r is in D, fo many times fliall E
be in E. Therefore E meafures both A and B ;
which confequently are not prime the one to the
other, contrary to the Hypothefis.

PROP. XXIV.
A, 9. B, 4. Numlers J, 2?, being the haft

C of all that have the fame propor-

D E - - tion with themy are prime the one

to the others.
If it be poflible, let A and Bhave a common
meafure C ; and let the fame meafure A by D,
a 9. mx. 7. and B by E ; a therefore CD— A, b and CEtt B.
b 17. 7. b Wherefore A. B D. E. But D and E are lelTer
than A and B, as being but parts of them.
Therefore A and B are not the leaft in their pro-
portion, againji the Hypothefis.

PROP. XXV.
A, 9. B,4. If two numbers Ji By be prime
C, 3. D - - the one to the other , the number C

tneafuring one of them A> Jball
be prime to the other number B.

For if you affirm any other D to meafure the
numbers B and C, a then D meafuring C does
a ii,a#.7.a]X0 meafure A , and confequently A and B are
not prime the one to>the other. Which is againji
the Hypothefis.
S PROP.

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EUCLIDE'.; Elements'.

*J7

PROP. XXVI.

A, 5. C, 8. // two numbers, A,B,b*-

B, }9 prime to any number C, the
a u £ . number alfo produced of them

F A, B (hall be prune to the fame

C

If it be poffible, let the number E be a com-

AB

mon meafure to AB, and C, andlet-g-bcrrV; aZ 9-ax* 7*

thence ABn EF; b wherefore alfo E. A B. F.^
But becaule A is prime to C, which Emeafu res, c z^ ym
c therefore E and A are prime to one another, d j ^m
and fo leaft in their own proportion, e and con- e zl% j9
fequently they muft meafure B and F } namely
F fliall meafure B, and A fhall meafure F.
Therefore feeing E measures both B and C,they
fhall not be prime to one another : Contrary
to the Hypothefis4

PROP. XXVII.

• ■

A, 4. B, 5. If two numbers A, B, be prime

Aq, 16. to one another, that alfo which is

D, 4. produced of one of them (Jq) jl)a%

be prime to the other B.
Take D = A $ therefore each of D, and A a 1. ax- 7
are prime to B. b wherefore AD or Aq is prime b 2.6. 7«
to B. lFlnch was to be dem.

#

PROP. XXVIII.

A, 5. C, 4. If two numbers Ay B, be. prime

B, 3. D, 2. to two numbers C, D, each to ei-
ABj$* CD,8. ther °f both, the numbers alfo

produced of them AB, CD fiall
le prime to one another.

For being A and B ajre prime to C,a therefore a 2£ ;.
fliaU AB alto be prime to the fame. And by the

y w

I58

b 26. 7.

7l5tf feventh Book of

fame reafon fhall AB be prime to D. h Therefore
AB is prime to CD. Which was to be dem.

PROP, XXIX.

A, j. B, z. If two numbers Ay B, be pihne
Aq, 9. Bq, 4. to one another , and each multiply-
Ac, 17. Be, 8. ing him/elf produce another number

* (Aq, and Bq;) then the numbers

produced of them (Aq, Eq) Jball be prime to one ano-
ther. And if the numbers given at jirft, A,B, multiply-
ing the faid produced number s(Aqybq)produce others
(Ac,Bc) thofe numbers alfo Jball be prime to one ano-
ther : And this pall ever happen about the extremes.

^7.7* For becaufe A is prime to B, a therefore Aq
lhall be prime to B. and Aq being prime to B,
a therefore Aq lhall alfo be prime toBq. Again,
becaufe A is as well prime to B and Bq. as Aq

b z8. 7. is to the faid B and Bq, b therefore fhall A x
Aq, that is, Ac, be prime to B x Bq, that is,
to Be ; Andfo forth of the reft.

PROP. XXX.
85 If two numbers AB9

A ........ B C I } D BC,be prime the one to

the other, then both ad.
ded together (AC) fhall be prime to either of them
• JBj Sc. And if both added together AC be prime to
any one of them AB, tbcAumbers alfo given in the
beginning AB, Bff^ftalt be prime to one another.
1. Hyp. Forlf you would have AC, AB to be
a il.ax.Jn compofed, let D be the common meafure : a
this fhall meafure the ieiidue BC : and there-
fore AB, BC, are not prime to one another j
which is againft the Hypothefis.

z. Hyp. AC, AB being taken for prime to one
another, let D be the common meafure of AB,
b ioukc.7. BC. b But feeing that meafures the whole AC,
therefore AC, AB, are not prime to one ano-
ther j contrary to tb§ Hypoihtfist

Google

EUGLIDE'j Elements] iy9

COYOU.

Hence, A number which being compounded
of two, is prime to one of them, is alio prime
to the other,

■

PROP. XXXI.

■

A, $• B,8. Every prime number A is prime toe-
very number By which it meafures not.
For if any common meafure does meafure
both A, B, a then A will not be a prime num-
ber 5 contrary to the Hypatbefis. a t tJefij^

PROP. XXXII.

A, 4. D, If two numbers A, B, multiplying

B, 6- E, 8. one another produce another Aby
AB, 24. an^ fome prime number D meafure

the number produced of them AB $
then Jhall it aljo meafure one of thofe numbers , A
§r By which were given at the beginning.
Suppofe the number D not to meafure the ntim*
AB

ber A, and let -~-=E. a then ABz=DE,iwhence a ^

D. A :: B. E. c But D is prime to A 5 d there- b J* 7-
fore D and A are the leaft in their proportion 5 c Wfr av*
e and confequently D meafures B as often as A J1' 7*
meafures E. Which was to be dem. d z J* 7*

€ 2f. ?.

PROP. XXXIII.

1

A, 12. Every compofed number A is meet*

B, 2* fared by fome prime number B9
Let one or more numbers a meafure A,of which

let the leaft be B 5 that fhall be a prime number : a i^defi.
For if it be faid to be compofed,then fome a lelfet
number lhall meafure it, b which fliall alfo Confe- K
quently meafureA. Wherefore Bis not the leaft D 11
of them which meafure A, contrary to the Hyp.

BR OP,

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x6o The fevtnth Book of

PROP. XXXIV.

A, p. Eve>y number J, is either a prime, or

vieafured by fovie prime number*
For A is either necelTarily a prime or a com-
pofed number. If it be a prime, 'tis that we
a 33* 7- affirm. If compoled, a then fome prime number
meafures it. Which was to be dem.

PROP. XXXV.

A, 6. B, 4* C/j 8. ft t v

D,z: H--I-K— -

*

i/bn? NUMf numbers foever A, jB, C, being given,
to find the leaft numbers Ey F9 G, that have the
fame proportion with them.
a 23. 7. If A, B, C, be prime to one another, a they
b j. 7. ' fhall be the leaft in their proportion. If they
be compofed) b let their greateft common mea-
fure be D, which let mealure them by E, F, G.
Thefe are then leaft in the proportion A, B. C.
c 9. djf.7. For DxE,F,G,f produces ABC, d therefore
d 17. 7. they are all in the fame proportion. But allow
other numbers H, I, K to be the leaft in the
e 2t. 7. fame proportion ; c which mail therefore equally
f 9. ax. 7. meafure ABC, namely by the number L. /there-
g 1. ax. 1. fore L x H,I,K, fhall produce A,B,C, g and con-
K 19- 7- fequently EDrrArHL h from whence E. H::
k fuppof. L. D. But E k cr K 5 I therefore LcrD, and fo
IzoJef.j, D is n. t the greateft common meafure of A,

B, C. Which is againfi the Hypothecs.

Lor oiu

Hence, The greateft common meafure of how
many numbers foever, does meaiure them by
the numbers which are leaft of all that have
the fame proportion with them. Wherebyap-
pears the vulgar method of reducing tractions
to the leaft terms-

i PROP.

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2*. * %

s

EUCLIDE'x Elements. 161

PROP. XXXVI.

Two numbers Icing given J, B, to find out the
leaft number which they meafure.
A, $. B, 4. 1. C*fe. If A and B be prime the

AB, 20, one to the other, AB is the num-

D 5er required. For it is manifeft

E — F that A and B meafure AB. If it

be poffible, let A and B meafure
fome other number D -73 AB, if you pleafe by
E, and F. a therefore AE - -I) - BF, bzna fo A.a 9- **.<f.
B :: F. E. But becaufe A and B c are prime the &
one to the other, d and fo leaft in their proper- k r9« 7*
tion, A fhall e equally meafure F as B does E.c
ButB. E/:: AB. AE (D) g Therefore AB fhall d 23-7-
alfo meafure D, which islefs than it felf. Which* Zl- 7-
is abfurd. » f 17. 7*

g loM.f.

A, 6. B, 4. F 2. Cafe. But if A and

C, D, 2; G--H--- B be compofed one to

AD, 12. another, /; let there be u 2 e 7

, , found C and D the 5>'/#

leaftinthefameproportfon.fethereforeAD" BQ k I0 -
and AD or BC fhall be the number fought for. y' '*

For it is /pkiri t^hit B and D do meafure AD 1 7# aXt *
pr BC. Conceive A and B to meafure F "3 AD,
iiamely A by G, and B by H. m therefore AG— m p# ax-m
F=BH. n whence A. B :: H. G 0 :: C. D. f and n T0' -
confequently C equally meafures H as D does G. 0 conar
But D. G q :: AD. AG (F.) therefore AD r mea- D Zl J- '
lures F, the greater the left, #7; is abfurd. *J f ^ ' ]

Corott. 1

, Hence, If two numbers multiply the leaft
that are in the fame proportion, the gi eater the
lets, and the lefs the greater, the leaft number
which they meafure fhall be produced.

r t prop:

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The ftvtktb Book of
PROP. XXXVII.

A, i. B, 3. If two numbers Jy B, meafure

E — 6 any number C, D, /Z^e num-
C F- ---D £er which they meafure E fiallai-

fo meafure the fame CD.
If you deny it, take E from CD as often as
you can, and leave FD -3 E. therefore feeing
A and B a meafure E, b and E meafures CF, c
likewife A and B will meafure CF. But a they

PROP. XXXVIIL

A, J. B, 4. C,6. Three numbers being given
D, ia. Jy Bs C, to find out the Uajt

which they meafure*
a Find D to be the leaft that two of them A
and B do meafure \ which if the third C do
alfo meafure, it is manifeft that D is the num-
ber fought for. But if C do not meafure D, Jtt
E be the leaft that C and D do meafure* E
fliall be the number required.
A, 2, B, j. C, 4. For it appears by the 11.
D, 6. E, 12. ax. 7. that A, B,C meaiure
F - - - E ; and it is eauly fliewn

that they meaiure no other
lefs than F, For if you affirm they do, b then D
meafures F, b and consequently E meaiure> the
fame F, the greater the lets. Which is abfurd.

Coroll.

Hence it appears, that if three numbers mea-
fure any number, the leaft alfo, wnich they
meafure, fliall meafure the bine.

PROP.

EUCLIDE'i Elements.

PROP. XXXlX.

A, 12. If any number B meafure a num-

B, 4. C, 3. her Ay the number meafured A, Jball

have a part C denominated of the
number meafuring £,

For becaufe £a -C9 b fliall A-BC. c there- f hyp.

A _ _ . . 7 j j c 7* ax* 7*

fore — B. Whch was to be devn '
C

s

PROP. XL*

A, i$. If a number A have any part

B, 3. C, 5. whatfoever if, /7;e number C, /iewi

n^ic/j ffo ^irf £ i* denominated,
Jball meafure the fame.

For being BC a = A,i thence^ — B. Fife* a &

tobedem. by.ax.f.

PROP. XLI.
i G,J2, To owf <i number G, w&iffi
H the leajt , contains the parts given

a Let G be found the leaft which the deno- a g ^
minators, 2, 3, 4, meafure \ b it is evident that ^ * "
G has the parts If it be poffible let H $y" '*

-5 G have the fame parts , r therefore 2, 3, 4, c 40. 7-
meafure H } and Lb G is not ihe leaft which
2, j? 4 meaiure : dgaiw/? conjlr.

•

The End of the feventh Book.

Li THE

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THE EIGHTH BOOK

OF

EUCLIDE', ELEMENTS.

PROP. I.

A, 8. B, 12. C, 18* D, 27.

E-F-Q—H—

IF there be divers numbers how many foever iri
continual proportion, ^,#,C,D, and their ex-
tremes A ,D, f rime to one another : then tlwfe
numbers AyByCJ), are the leaft of all numbers
that have the fame proportion with them.
For, if it be pomble, let there be as many
others E, F, G, H, lefs than A, B, C, D, and in
the fame proportion with them, a Therefore of
equality A, D :: E. H. and confeqtiently A and
D are prime numbers, b and fo the leaft in their
proportion, c equally meafuring E and H,which
are lefs than themfelves. Which is abfutd*

PROP- II.

»

*

A, *• B,
Aq, a. AB, 6. Bq, 9*
Ac, 8. AqB, iz, ABq, 18. Be, 27.

To find out the leaft numbers continually propot-
tionaly as many as J}) all be required, in the propot-
tion given of A to B.

Let A and B be the leaft in the proportion
given 5 then Aq, AB, Bq, fhall be three laft in
the fame continual proportion that A is to B.

For

Digitized by Google

- EUCLIDE'f Elements'. I6f

For A A. AB a ;: A. B a :: AB. BB. Likewifea 17. 7,
becaufe A and B are b prime to one another, cb 14.7.
fhall Aq, Bq, be alfo prime to one another, dc 2,9. 7.
and fo Aq, AB, Bq, are if the leaft in the pro-d I, 8,
portion or A to B.

Moreover, I fay Ac, AqB, ABq, Be, are the
four leaft in the proportion of A to B. For AqA.
AqB e -.: A. B. e :: ABA (AqB.) ABB. e and A. e 17. 7.
B :: ABq. BBq (Be.) Therefore fince Ac,and Be, f 29. 7.
are /"prime to one another, likewife^ mail Ac, g 1. 8.
AqB, ABq, Be be the four leaft rf in the pro-
portion of A to B. In the fame manner may you
find out as many proportional numbers as you
pleafe. MHnch was to be done.

CorolL

1. Hence/ If three numbers, being the leaft,
m are proportional, their extremes fhallbe fquaresj

if four, cubes,

z.How many extremes proportional foeverthere
be,being by this prop, found to be the leaft in the
given proportion, they are prime to one another.

2. Two numbers, beingthe leaft in the given
proportion, do meafure all the mean numbers
whatfoever of the leaft in the fame proportion ;

• becaufe they arite from the multiplication of
them into certain other numbers.

4. Hence alfo it appears by the conftruftion,
that the feries of numbers 1, A,Aq,Ac ; i,B,Bq.
Be; Ac, AqB, ABq, Be confifts of an equal multi-
tude of numbers; and confequently, the extreme
numbcrsof how many foeverthe leaft continually
proportionals are the laftof as many other conti-
nually proportionals from a unite, as the extreme
Ac,Bc,of the continually proportionals Ac,AqB.
ABq, Be, are the leaft of as many proportionals
from a unite, 1, A, Aq, Ac ; and i,B,Bq, Be.

5. 1, A, Aq, Ac ; and B, B A, BAq ; and Bq,
ABq, are rf in the proportion of 1 to A. Alfo
B, Bq, Be ; and A, AB, ABq ; and Aq, AqB are
Jf in the proportion of r to B.

X- j PROP,

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I £6 The eighth Book of

i

PROP. III.

A, 8. B,u. C, 18. D,z8. If there be numbers

continually proportion
vaU how manyfoevcY) J>ByCyD, being alfo the leaji
of all that have the fame proportion with them I
then extremes Jy D, are prime to one another.
a 2. 8. For it there be a found as many numbers the
leaft in the proportion of A to B, they fliall be
no other than A, B, C, D ; therefore, by the fe-
cond Coroll.oi the precedent Prop, the extremes
A and D are prime to one another. Which was
to he dcm.

»

PROP. IV.

A, 6. B, 5. C, 4. D, 3. Proportions how ma-
il, 4. F, Z4. E, 2.0. G, 1 f. vy foever being given*

I . . K - ' - L in the leaft number s{A.

toB, andCtoD)) to
find out the leaft numbers continually proportional
in the proportions given.
a 56. 7. a Find out E the leaft number which B and C
b l-poftj. do meafure ; and let B meafure E b as often as A
does another F, viz. by the lame number H. b
Alfo let C meafure the faid E as often as D mca-
fures another G. then F, E, G, mall be the leaft
c 9. ax. 7. in the proportions given. For AH c = F, .and
di8.7- BC* :-Ej d therefore A. B :: AH.BH^ ::F.R
e 7. 5, In like manner C. D :? E. G ; therefore I , E, Cj,
are continually proportional in the proportions
given. And they are moreover the leaft in the
laid proportions : For conceive other numbers,
f 11. 7. 1,K,L, to be the leaft 5 / then A and B lrmft e-
qually meafure I and K, / and C and D likewUe
K and L ; and fo B and C meafure the fame K.
g 37-7- £ Wherefore alfo E meafures the fame number
K, which is lefs than it felf. Which is abfurd.

A,<5.

♦

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' EUCLIDE'i Element i. 167

A, 6. B,5. C,4. D,$. E, 5. F,7-
H,24. G,2o. I> is-
But three proportions being given, A to B, C
to D, and E to F; find out as before thiee num-
bers H, G, I, the leaft continually in the propor-
tions of A to B, and C to D. Then if E mealures
I, h take another number K which maybe equal- h l»poft.j.
ly meafur'd by F ; and thofe four numbers H,G,
I, K, lhall be continually the leaft in the given
proportions ; which we need go no other way
to prove than we did in the firft part.

A,& B,*. C,4. D,$. E,z. F,7-

H, 24. G,29-
M,48. U40. K^o. N,ioj.
If E do not meafuie I,let K be the leaft which
E and I do meafure ; and as often as I meafures
X, let G as often meafure L, and H alfo M. lb
likewife let F meafure N as often as E meafures
K. The four numbers, M, L, K, N lhall be leaft,
continually in the given proportions 5 which
we may demonftrate as before.

a1

P R Q,P. V.

C, 4. E, 3. Plane numbers CD, p

D, 6. F,i6. £D,i8. EF* are in that propor-
CD^.EF,48. Uontoone another wlpch

^ iscompofed of then fides.

For becaufe CD- ED a :: C.E \ a and ED. EF :: * *7« 7*

D. F. 8c b S5 -»• 2*thenfhallbethepvo- b

EF ED E en. $•

CD C D
portion ^ = - -h Winch was to le dem.

PROP. VI.

A, 16. B, 24. C, 36. D, 54. E,8i. If there he
F,4. G, 6. H, 9. number scouts

v mtally propor-

tional how many foever> Ay 2?, C, D, £, and the ft ft
A do not meafure the fecond B, neither Jball any of
the other meafure wy on* of the rejl9

L 4 **e^

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1 68 The eighth Book of «

a zo Jef.j. Becaufe A does not meafure B, a neither fhall

any one meafure that which next follows } be-
b 1 5. 7. ing A.B :: B.C :: C-D.&c. b Take three numbers,

F, G, H, the leaft in the proportion of A to B.
therefore fince A does not meafure B, a neither

c <. ax. 7 ^all F meafure G. c therefore F is not a unit. But
d }1 J. f and H are prime to one another; and fo,<* be-
e 14. 7, *n8 °f equality A. C::F.H. and F does not mea-
' ' fure H, a neither fhall A meafure C ; and con-
fequently neither fhall B meafure D, nor C mea-
fure E,gfr. becaufe A. C e :: B. D e :: C.E,SrV.
In like manner four or five numbers being taken
the leaft in the proportion of A to B, it will ap-
pear that A does not meafure D andE} nor does
B meafure E and F, &c. Wherefore none of them
fhall meafure any other. Winch was to be denu

PROP. VII.
A, B, 6. C, 12. D, 24. E, 48.
If there be numbers continually proportional how
many foever ^,5,C,D, £, and the firft A meafure the
laft Ey it Jhall alfo meafure the fecond £.
3 6, 7. If you deny that A meafures B, a then neither
fhall it meafure E } Winch is contrary to the Hyp.

PROP. VIII.
' A,24. C,5& D,54- B,8i. If between two

G, 8. H,I2. I,l8. K,27. numbers J, By there
E,32. L,48. M,72. F,io8. fall mean proportional

numbers in continual
proportion C, D } as many mean continually propor-
tional numbers as fall between them, fo many alfo
mean continually proportional numbers flail fall be-
tween two other numbers £, F, which have the fame
proportion with them. (L, M.)
a 3 S- ?• a Take G,H,I,K, the leaft fr in the proportion
t>M-7- of A to b of equality fhall G.K:: A.Bc::E.F.
c bit' But G, and K d are prime to one another, e
d 3. 8. Wherefore G meafures E as often as K does F.
e u. 7- Let H meafure L, and I likewife M by the fame
f conjlr. number. / therefore E, L, M, F, are in fuch pro-
portion as G,H,I>K, that is as A,B,C,D. which
v *ra* to be dem. PROP*

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EUCLIDE'* Elements,

PROP. IX.

i. s If two numbers Jy B

E,2. F,J. be prime to one another,

G,4. H,6. and mean numbers in

A,8. C,iz. D,i8,B,i7. continual proportion C,D

fall between them ; as ma*
fiy mean numbers in continual proportion as fall bc~
tween tbem9 fo many means alfo (E, G ; and F. I)
Jball fall in continual proportion between either of
them and a unity.

ft is evident that r , E, G, A, and i, F, I, Bf
are ~ , and as many as A, C, D, B, namely by
the 4. Coroll. z. 8* Which was to be dem.

PROP. X.

A*8. I,iz. K,i8. B,Z7» If between two number?
E,4. DF,6. G.9. J,B, and a unit, numbers
D,i. F,$. continually proportional

1. (E,D,andF,G,)<foM

how many mean numbers
in continual proportion fall between either of them
and a unit, fo many means alfo JbaU fall in conti-
nual proportion between them, J, K.

For E, DF, G, and A, DqF (I) DG, (K,) B,
arc t: by 2, 8. therefore, Qfc.

PROP. XL

A,2. B,3. Between two fquare numbers

Aq,4« AB,6. Bq,9« Aq,Bq, there is one mean pro-
portional number JB : and
the fquare Aq to the fquare Bq is in double propor-
tion of that of the fide A to the fide B*

a It is manifeft that Aq,AB,Bq, are i and a ,7# 7;

confequently all

• <

PROP.

T9

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*

I70 The eighth Book of

prop. xir.

Ac,27. AqB,}6, ABq,48. Bc,64- Between two
A, j. B, 4. rwie numbers,

Aq, 9 AB, 12. Bq, 16. Ac, Be, there

are two mean

-proportional numbers AyB, ABq: and the cube Ac
is to the cube Be in treble proportion of that in
which the fide A is to the Jide B.
a 2. 8- a F°r Ac, AqB,ABq, Be, are ~ in the propor.

b ioJef.$. t ;0I1 of A to B; b and therefore -g^ — g« trebly.
JPWci to be dem.

■

PROP. XIII.

A, 2. B, 4. C,«.
Aq,4» AB,8. Bq, 16. BC,j2. Cq,64.
Ac,8.AqB,i6.ABq,j2.Bc,€4. BqC,i28.B(Jq,256.

(Cc,ji2.

If there be numbers in continual propor tion bow
many foever A,B,C\ and every oftoem viultiply'ing
it felf produce certain numbers \ the numbers produ-
ced of them Aq, Bq, Cq, Jball be proportional : And
if the number firft given A, B, C, multiplying their
prpdufts Aq, Bq, Cq, produce other numbers, Ac, Be*
Cc, they alfo pall be proportional j and this Jball
ever happen to the extremes.
a 2. 8. For Aq, AB, Bq, BC, Cq 0 are t: £ therefore
b 14. 7. °f equality Aq.Bq:: Bq.Cq. Which was to be dan.

a Alfo Ac, AqB, ABq, Be, BqC, BCq, Cc, are
*f?; b therefore likewite of equality Ac. Be ::
Be. Cc. Winch was to be dem,

PROP. XIV.
Aq>4. AB,I2. Bq,}6. // a fquare number Aq
A, 2. B, 6. meafure a fquare number

Bq, the fide alfo of the one
{A) JhaU meafure the fide of the other (B:) and if the
fide of one fquare A meafure the fide of another B,the
fquare Aq Jball likewifc meafure the fquare Bq.

Digitized by Google

EUCLIDE'j Elements. lyx

i. Hyp. For Aq . AB a :: AB. Bq. therefore fee-a2.gfi j.8
ing by the Hypothefis Aq meafures Bq, b itb 7. 8.
IhaiJ meafure alfo AB. But Aq. AB :: A. B. cczoJef.j*
therefore A mealures B. Winch was to he dcm.

z Hyp. A meafures B. c therefore Aq fhall.as
well meafure AB, c as ABmeafures Bq;aandcon-d ii.ax.f.
fequently Aq meafures Bq. Which was to be dew.

PROP. XV.

A, 2. B,6. If a cube num~

Ac>8.AqB,24.ABq,72,.Bc>2i<5. ber Ac meafures a

cube number Be y
then the fide of the one (A) fbaU meafure the fide
of the other (B:) And if the fide A of one cube Ac
meafure the fide B of the other Bcy alfo the cube
Ac Jball meafure the cube Be.

r Hyp. For Ac,AqB, ABq, Be a are frvthere- ?2:^I2#8
fore Ac, £meafuring the extreme Be. fhall alfo"
c meafure the fecond AqB. But Ac. Aq. B .*; A.^ 7- &
B. d therefore A fhall alfo meafure B. dzoJefr.

z. Hyp. A meafures B 9 d therefore Ac mea-
fures AqB, which alfo meafures ABq, and that c *l*ax'7*
Be 5 e therefore Ac fhall meafure Be, Winch
was to be dem.

PROP. XVI.

A q, 4. B, 9. If a f juare number Aq do not mea-
Aq,i6.Bq,8i . pure a fquare number Bq^neither Jball

the fide of the one A meafure the fide
of the other B : And if A the fide of the one fquare
Aq do not meafure B the fide of the other Bq, nei-
ther Jball the fquare Aq meafure the fquare Bq.

i.HyprVoi if you affirm that A meafures B, a a *4» 8«
then Aq alfo fhall meafure Bq. againjl the Hyp.

z. Hyp. If you maintain Aq to meafure Bq ;
a then lfkewife A fhall meafure B. contrary to
the Hypothefis.

P&OP.

1

1

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J7? The eighth Book of

PROP. XVII.

f

A,2. B,j. If a cube number Ac do not mea-
Ac,8. Bc,27. /ur* a cube' number Be , neither

Jball the fide of one Ay meafure the
fide of the other B ; And if A the fide of one cube
Ac do not meafure B the fide of the other ik, neither
Jball the cube Ac meafure the cube Be.
IS* 8* i. Hyp. Let A meafure B; 4 then Ac fhall
rneafure Be. agamfi the Hyp.

i. Hyp. Let Ac meafure Be ; then A fhall ipea~
fure B j which is alfo againfi the Hypothefis.

{ PROP. XVIII.

C> &,D, 2. Between two like plane num."

CI), 12. hers CD and EF there is one

E,J. I**,}. DE,i8. mean proportional number DE :
EF, 27. And the plane CD is to the

plane EF in double proportion
of that which the fide C has to the homologous fide
(or of like proportion) £.

irv^f < ^h. £F- c Wherefore the proportion of CD to
ictffr.5.EF is double to that of CD to DE, that is, to
the proportion of C to E, or D to F.

m

CorolL

Hence it is apparent, that between two like
plane numbers there falls one mean proportional
in the proportion of the homologous fides.

PROP,

, 't

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EUCLIDE'i Elements.

PROP. XIX.

i

t

CDE, 30. DEF,6o. FGE, 120. s?QU9Uo;
CD, 6. DF,i2. FG,i+.
C,z. D,$. E, $. F,?. G,<S. H,io.

•

Between two like/olid numbers CDE, FGH, there
are two wean proportional numbers DFE> FGE.
And the f olid CDE is to the/olid FGHjn treble pro-
portion of that which the homologous fide C has to
the homologous fide F.

Whereas by the * hyp. CD :: F.G, 8? D.E •* nJefa.
G. H. therefore a by inverfion fhall Q F D. a i;« 7.
G :; a E.H. But CD. DF b :: C. F, and DF. FG b 17. 7.
* :; D. G s c wherefore CD. DF ..• EfF. FG :: c 11. J.
E.H. J and accordingly CDE. DFE.\- DFE. d 17.7.
FGE .v E. H .v FGE. FGH. Therefa/e between
CDE. FGH, fall two mean proportionals DFE,
FGE. e And fo it is plain that the proportion c iQJe}&~
of CDE to FGH is treble to that of CDE to
FDE, or C to F. Which was to be dem.

CoroU.

Hereby it is manifeft, that between two like
folid numbers there fall two mean proportionals
in the proportion of the homologous tides.

PROP. XX.

A, 12, S,i8. B, 27. If between two numbers
D,2. E,j, F,6. G,9. Jy B9 there fall one mean

proportionalnumberCithefe
vumbers J, By are like plane numbers.

[a Take D and E the leaft in the proportion of A a 55. 7*
to C, or C to B. then D meafures A equally as E
does C,itf z.by the lame number F \ b alfo D equally b 21 • 7.
meafures C, as E does B, viz. by the fame number
G. c Therefore DF=: A, and EG^B. <famicon- q p.ax.j.
fequently A and B are plane numbers. But becaufe d i6.def.jj
iF^CtfszDG, e fhall D. E Ft G. and alter- e 19. 7.

nately

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Xnj^ The eighth Book of

Fii.^/.7.nate^y D.F :: ^-G- / Therefore the plane numbers
A and B are alfo like. Which was to be dem.

PROP. XXI;

A3i6. 0,2,4. D, 36. B, 54. If between two
E, 4. F,6. G, 9. numbers J, £ there

H,2. P,z.M,4.K,5.L,^.N,6. fall two mean propor-
tional numbers C,D }
thofe numbers J, B are like folid numbers.
a 2. 8. * Take E, F, G, the leaft 45 in the proportidn
b 10.8. of A to C. b then E and G are like plane num-
bers: let the fides of this be Hand P, and of that
czi.def.7 K and L. c therefore H.K::P.L.\ 'dE. F. But E,
d com 8.8. F, G, do e equally meafure A, C, D, viz. by the
e 21. 7. lame number M. and iikewife the iaid numbers
E,F,G, do equally meafure the numbers C,B,D,
viz. by the lame number N. /Thcreiore A EM
f 9. ax. 7.^HPM, / and Bzr GN =: KLN ±g and Lb A and
Cl7ufof.7,B are folid numbers. But for that C /—FM, and
h 17.7. D f = FN, therefore lhall M. N i :; FM. FN k
k 7. $. .*.* CD / :: E.F H.K P. L. m wherefore A and
1 conjlr. B are like folid numbers. Winch was to be dem*
jnu.def.j. Lemma.

AE, BF, CG, DH, If proportional numbers J9
fc, A, B, C, D, C, meafure proportional

E, F, G, H. numbers AE, Bt\ CG, DHy by

the numbers £, F, G, //, r/;e/e
numbers (£, F, G, /f,) ./fta// £0 proportional.

For being AEDH ar= BFUJ, a and ADrrBC,
a 19. /• AEDH BFCG

bi.ax. 7. J thence will — — — — - c that is, EH=FG.

£ Q. ax. 7» AU !5L»

J Therefore E.F G.H. Which was to be dem.

Cor oil.

Ba B B

d i$Jefr. Hence^S.= - dFoi t.B;;B.Bq^and 1. A.v

c lem.prec » A. Aq.e theref.-* B.v^ d theref. — -= -

A. A Aq Aq A A

B B Be
In like manner — < — =r , and fo of the 1 eft.

A£ AC ACC

PROP,

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EUCLIDE'j Elements'. i7j

PROP. XXIL
Aq, B, C. If three numbers Aq> B9 C he
4. 8. 16. continually proportional, and the
jirft Aq a fquare, the third C
fiall alfo le a [quart.

B el

For becaufe Aq C tfnBq, I thence is C = ^3 a IO- 1%

B B b ax-7-

c~Q.~£ ; But it is plain that «-~is a number^ be- c cor. of the
tT A lem.prec.

caufe p-orC isanumber. Theref. ifthree,^, <* ™*
Aq 14. 8.

PROP. XXIII.
Ac, B, C, D, If four numbers Ac, B,C>D9
.8, 12, 18, 27. be continually proportional, and

the firjt of them Ac a cube9 the
fourth alfo D fiall he a cube.

BC

For becaufe Ac Dtf=BC, £ therefore D~ A a 19. 7.

■**c |^ tdx^ 7^

C3=^x c j that is (becaufe AcC = d Bq, and C™*Q£

^, Bq n B Bq Be >, B d 20. 7.

Ac Ac Ac Acc Ac,

B . . , r Be e 15. a
But it is evident c thatAc is a number, becaufe ^

or D is fuppofed a number. Therefore if four ,
numbers, gjv,

PROP. XXIV.
A, 16. 24. B, 36. If two nimbers AyB, le in the
C,4. 6. D,9. fame proportion one to another ,

that a fauare number C ts to ct
fauare number D, andthefirjt Abe a Jquare number,
tlie fecond alfo £ /ball be a fquare number.

Between C and D being legate irum1 cs,*and * 8. 8*
fo beiween A and B having the fame proportion a **• 8.
a falisone mean proportional, Therefore b ucing 3 hyp.

A

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Ij6 The eighth Book of

C it. 8. A is a fquare number, c B alfo fliall be a fquare
x number. Which was to he dem.

Coroll.

*. Hence, If there be two like numbers AB,
GD (A. B:: CD) and thefirft AB be a fquare;

* ©the fecond alfo CD fhall be a fquare,

*n.fcr i» ^ For AB CD Aq Cq

2. From hence it appears, that the proportion
of any fquare immber to any other not fquare,
cannot poflibly be declared in two fquare num-
bers. Whence it cannot be Q. ;; i. 2, ftbr i«
i Q. Q, &e.

PROP. XXV.

0,64.96. 144. D, 216. If two numberi A,B, ht

• A, 8. 12. 18. B, 27. in *Ae fame proportion one

to another, that a cuhe
number Cis to a cuhe number D, the firll of them
A being a cube number \ the fecond S JbaU likt-

Ijg g ' a Between the cube numbers C and D, b
cfap* m* t° between A and B having the fame pro-
d 2*. 8. P01**0"* two mean proportionals 5 therefore

c iecaufe A is a cube, d fliall B be a cube alfo*

Winch was to be dem.
/ Coroll.

1. Hence, If there be two numbers ABC,

DEF (A. B D. E, and B. C :: E. F;) and the

firft ABC be a cube, the fecond DEF fliall be a

cube alfo.

gI2>GM9» 1. It is* perfpicuous from hence, that the
proportion of any cube number to any othet
number not a cube cannot be found in two
cube numbers.

PROP,

1

Digitized by Google

EVCUDE's Elements'. 177

t R O P. XXVI.

A,20. C,jo, B,45* Like plane numbers AyB±
Dj^. F,9» dre in the fame proportion one

to another y that a fquarenum-
her is in to a /quart number.

Between A and B a falls one flnean propor-a
tional number O; b take three nwnbers D,E,F,b ju 8.
the leaft *?i in the proportion of A to C, the
extremes D, F, b lhall be fuuare numbers. But
of equality A. Be :: DJF. therefore A. B :: O.Q.c , A 7
Winch was to be dbm. ^Cff7,

p it o p. xivn.

0

A, 1 6. 0,24. D,3<$. B,54« Like /a/ti numbers
£,8." F,i2. G,i8. H,27. arc ia the/ame

proportion one to ano~
thery that a cube number is in to a cube number.

a Between A and B fall two mean proportio-a 0 o
nal numbers, naribely C andD: b take fourh '
numbers E, F, G, H the leaft * iA the proper- 0 1# 0#
tion of A to C \ b the extremes E, H, are cube ^ , A m
numbers. But A. B c :: E. H :: C. C. **• 7-

irjy ht..itm.

SchoL

1. From hen<Se is inferred, thdt no numbers in See C7*~
proportion fuperparticular, fuperbipartient, or??*?,
double, 6t any other manifold proportion not
denominated from a fquare, number, are like

jplane numbers,

2. Likewife, that neither any two prime
numbers, nor any two numbers prime one to
another, not being fquares, can be tiki plane
numbers. *

The End of the eighth Book.

*

* t 9

M • THE

Digitized

a 17. 7-

THE NINTH BOOK •

OF

EUCLIDE'i ELEMENTS.

'• PROP. I.

A,6. B.S4.

Aq, 36. 108. • AB, 324.

* 1

Ptwo like plane numbers A, E multiplying one
another, produce a number AB, the number
produced AB Jbatt be a fouare number.
For A. B a :: Aq. AB ; wherefore finCe

5 tg 3 one mean proportional b falls between A

c g g and B, c likcwife one mean proportional number
fhall fall between Aq and AB : therefore being

d £2. 8. ^9 *s a ^uare number, d the third AB

s * ' fhall be a fquare number too. Winch was to be
dcmonjlratedi

Or thus. Let ab, cd, be like plane numbers ;
% 19. 7. namely a. b :: cd. x therefore ad— be. and fo like-
y 1. ax. j. wife ab cd, or ad be, = ad ad = Q.: ad.

PROP. IL

A, 6. B, $4. If two 7iumbers A, B, multi-

Aq, 36. AB,324* plying one another, produce a

fquare number AB, tuofenum*

hers A, B are like plane numbers.
* 17. 7. For A. B a :: Aq. AB , wherefore being be-
b 11. 8. tween Aq, AB, b there falls one mean propor-
c 8. 8. # tional number, c likewife one mean (hall fall be-
d 20. 8, tween A and B. d therefore A and B are like

planes. Wl>icb was f be dm.

1 PROP.

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EUCLIDE'; ElemMA irf

prop, m

A, 2. Ac, 8. Acc, 64. If a cube number M

multiplying it f elf tiro*
jiucc a number Acc. the number produced Acc Jball
be a cube number.

For 1. A a :: A. Aq b -.: Aq. Ac. therefore be- * 7*
tweeni and Ac fall two mean proportionals. But^ *?• 7-
1. Ac a :: Ac. Acc. c therefore between Ac and c 8* 8.
Acc,' fall alfo two mean proportionals : and fo ^ *J«8»"
by confequence feeing Ac is a Cube, d Acc fhall
be a cube alfo. Which was to be dem.

Or thus; aaa(Ac) multiplied into itfelfmakei
aaaaaa (Acc;) this is a cube, whofe fide is aa.

PROP. IV.

» .

Ac, 8. Be, 27. If a cube number Ac mnl-
Acc,64. AcBc,2i6. tiplying a cube numhr Be9

produce a number AcBc> the
produced number AcBc Jball be a cube.

For Ac. Be a :: Acc. AcBc. But between Ac a lf* g
and Be b two mean proportional numbers fall ; ^ Ir V>
c therefore there fall as many between Acc and ^\
AcBc. So that whereas Acc is a cube number, i ^ 1 J- **• *
AcBc fhall be fuch alfo. Winch was to be dem.

Or thus. AcBcrraaabbb (ababab) se C : ab.

PROP. V.

• «

Ac, 8. B, 27. If a cube number Ac rriulr
Acc,64- AcB,2il. tiplying a number B produce

a cube number AcBy the
number multiplied BJbaH alfo be a cube.

For Acc. AcB. a :; Ac. B. But between Acc a 17. 7^ „"
and AcB b fall two mean proportionals; c there- b 12. 8.
fore alfo as many fhall fall between Ac and B. c 8. 8.
whence Ac being a cube number, d B fhall be a d 2;. &
tube number top. jnich was to fo dem. '

P^OP.

Digitized

1S0 The ninth Book of

PROP, VI.
A,8. Aq,<4» Ac, $11. If a mmher J multi-

plying it felf poduce a
iitjUfisa '

cube Aq, that number A itjelfis a cube.
a hyp. For becaufe Aq a is a cube, and AqA (Ac) b
bi9.A/.7.alfoacube; therefore c fhall A be a cube. Which
c $. ?. was to be dem.

PROP. VIL
A, 6. B,ii. AB, 66. If a compofed number A
D, 2. E, $. multiplying any number B,

produce a number AB, the
number produced AB pall be a folid number.
a f : defa. Being A is a compofed number, a fome other
k J'j/fn number D meafures it, conceive by E. b there-

c?i dlf T forc A = DE : c whence DEB = AB is * folid
7 ;v# number. Which was to be dem.

PROP. VIII.
i.a9];a'99aS*7» a4,8r. a',*^. af,7i9.
If from a unit there be numbers continually pro*
portiotial how many foever (i.a, a*, a*, a4, &c)
the tlnrd number from a unit a1 is a fquare number ^
end fo art all forward, leaving one between (a4, a6,
a8, &c) But the fourth a' is a cube number $ andfo
tire all forward, leaving two between (a*, a9, &c)
2Hte feventh aljo a#, ix fctfi a cube number and a
ftuare $ and fo arc all forward, leaving five between
(a",a",&c.)

For 1. a* = Q. a. and a4 = aaaa = Q. aa.
and a6 = aaaaaa — Q. aaa, &c.

1. a' = aaa = C. a. and a6 c= aaaaaa C.
aa and aaaaaaaaa = C. aaa, &c.
, 3 • a c = aaaaaa =r C. aa rrr Q. aaa. therefore, &c.

* Or according to Euclidz\ Becaufe j; a

10*7- £ fhall aJ=Q; a. therefore feeingaa,aSa4,are fr9
c ia. 8. f the third a4 fhall be a fquare number , And fo
, ^ likewife a', a8, &c. Alfo becaufe 1. a d :; a1, a1,
a 2,3. 0. therefore fhall a* £ = a* xar C:a. there-
fore the fourth from a3, namely a*, fhall like-
wile be a cube, &c. and confequently ad is both
a cube and a fquare number, &c.

PROP.

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EUCLIDE'f Efewvts. 181

» •

PROP. IX.

i.a,8.a>,64.a,1$i^a4i4°9^ f&tre fti

numbers how

many foever continually proportional (i. a, a?, a*,
&c.) rt?^ number following the unit (a) 4
/fzune $ /7;«i rt<7 the rejt, a2, a', a4, 8cc. Jball be
fquares too. But if the number next the unit (a) be
a cube, then all the following numbers a', a1, a4f
&c. Jball be cuke numbers.

i. Hyp. For a:, a4, a", &c. are fquare numbers
by the prec. Prop, alfo being a is taken to be a
fquare, a therefore the third a* fhall be a fquare, a 22. 8;
and likewife a% a7, 8cc. andfo all.

z. Hyp. a is taken to be a cube, b therefore a4, b 2?. 8.
a7,al° are cubes: but by the prec. a\ a', a9, c 20. 7.
&c. are cubes: laftly,becaufe i.a :: a. aa. cthere» d 3. 9.
fore lhall a1 -r Q: a. but a cube multiplied intoc 23, £«|
it felf d produces a cube , there foie a* is a cube,
e and confequently the fourth from it a5, and in
like manner a1, a1 &c. are cubes, therefore all.
' Winch was to be dem.

Peradventure more clearly thus, let b be the
fide of the fquare number a, and fo the feries a9
al,a*,a4,&c. will be otherwife expreiTed, thus,
bb, b4, b% b8,&c. It is evident that all thefe
numbers are fquares, and may be thus expreffed,
Q? b, Q: bb: Q. bbb, Qj bbbb, &c.

In like manner, if b be the fide of the cube a,
the feries may be expreffed thus, b3, b*, b*,b,a.
&c or C: b,C: b2,C:b»,C: b4, &c.

PROP. X.
1, a,a% a', a4, a*, a6, Iff)om a unit there be
I, z, 4, 8, 16, 32,64, numbers how many foever

continually proportional
(i,a, a-, aJ, &c) and the number next the unit
(aj be not a fquare number \ then is none of the rejl
following a fquare number, excepting a1 the third
from the unit) and fo all forward, leaving one be-

M 3 tween

Google

Tie ninth Book of

tween(**,**&*,8cc.)But if thatfywhichisnext after
the unity be not a cube number* neither is any other
qf the following numbers a cube>faving a ; tfo fourth
from the unit, and fo of all forward leaving two be-
tween, a% a9, a'2, &c.

i. Hyp. For if it be poffible, let a* be a fquare
a hyp. jnumber \ therefore becaufe a. a* a :: a4, a' , and
hfuppofSc by inverfion a*. a4 :; a5, a 'r and alfo aT and a4
g p. are £ fquare numbers, and the firft a1 a fquare,
c 24. 8. f therefore a fhall be likewife a fquare j contrary
to the Hypothecs.

z. If it mav be, let a4 be % cube 9 being
^ of equality ^ a* a. a* and inverfely a*. a4
e 5 j. 8. a$. a 5 and alfo being a5 and a4 are cubes,
and the firft a? a cube } e therefore a fhall be a
pube alfo i againft the Hyp.

PROP. XL

d M. 7-

>

y. a, a% a^,a4, a$, a6. Jjf there be numbers
J, 3, 9, 17, 8r, 143, 719. how many foever in con-
tinual proportion from
a unit (1, a, a2, a$, &c.) the lejs meafmeth the

greater by feme one of them that are amongft the

proportional numbers.

aa aaa

a $.ax. 7. Becaufe j. a a. aa. 4 therefore — ^ a = —

*czoJef.7 " a aa*

|4 - - aaa
° f 4» /• Alfo becaufe 1Mb :: a.aaa.t;therefore - - = aa =

a

2- ,&c.Laftly becaufe 1 .a$ a^therefore
S3, a 3

a • -? aj

CorolL

Hence, If a number that meafures any one of
proportional numbers, be not orie of the faid
lumbers, neither (hall the number by which it
jneafuresthe laid proportional numbers, be one
fcf them,

1 PROPc

Digitized by Google

EUCLIDE'i Elements. i$}

11 ,

PROP. XII.

I, a, az, a; , z± If there be numbers how many
1,6,36,116,1296. foever in continual froportion
B. }. from a unit) (r, a, ai, a$, a4«)

whatfoever prime numbers B
meafure the laft ad, she fame (B) Jball alfo meafure
the number (a) which follows next after the unit.

If you fay B docs not meafure a, a then B is* ?*• 7-

f>rime to a ^ * and alfo B is prime to ai , c and& ^7- 7 •
o confequently to a.4, which is fuppofed to c *o. 7*
meafure. Which i* abfurd.

Coroll,

1. Therefore every prime number that mea-
sures the latt, dyes alfo meafure all thofe other
numbers that precede the laft.

z. If any number not meafuring that next to
the unit, does yet meafure the Jalt, it is a com
pofed number.

2. If the number next to the unit be a prime,
no other prime number fhall meafure the laft. x

■

PROP. XIII.

I, a. az, a}, a4, If from q unit be num-
l» 5? *5> 125,625. lers in continual propor-
H--G--F--E-- tion how many foever (a,

ai, a}. &c.) aiti that af-
ter the unit (a) a prime ; then Jball no otber meafure
the greateft number, but thofe which are amongfl
the /aid proportional numbers.

If it be poflible, let fome other E meafure a4,
viz. by F, a then F fhall be fome other befide a, a ttw.u.c.
ai,a$. But becaufe E meafuring a4,does not mea-b ixor.iu
fure a, b therefore £ fhall be a compofed num-o.
ber, c and fome prime number meafure ityi which c $3.7.
does confequently meafure 34, e and fo is no other d iiuw.7.
than a, therefore a meafures E. So alfo may Fc $.coj.I2.
be (hewn to be a compofed number, meafuring a^, 9,

M4 .

Digitized

184 The ninth Book of

meafuring a^, and fo that a meafures F. There-
f 9* ax. 7. fore feeing EF f~ 34 = a x a$, g fhall a. E :: F.
g ip- 7- a?. Confequently, whereas a meafures E, b like^
hzoJef.y. wife F fliall equally meafure a$, viz. by the
ka>r.i 1.9. fame numberG. k Norftiall Gbea.oraz. there-
fore, as before, G is a compofed number, and %
meafures it. Wherefore being that FG /— aj
= a 2 x a £ fliall a. F G a2. and fo becaufe A
meafures F, h G fhall equally meafure az. viz.
by the fame number H, which is not a. There-
1 Zljef.i. ^ore being GH rr az =r- aa, / thence H. a .v a.
m zo. <fe/.^* and becaufe a meafures G (as before) m H
- * J#alfo fhall gieafure a, which is a prime nun*-
ber. Which u impojible.

PROP. XIV.

■

A, jo. If certain prime numbers Cf

B,z. C,g. D,$. Dy do meafure the leaft number

E - - F J, no other prime number E Jbatt

meafure the fame, befides thofs
that meafured it at firft.

a * **" 9# If it is poffible,let ~ be = F. a then A = EF.

b I2" 7* b therefore every of the prime numbers B,C, D,
meafures one of thofe E, F, not E, which is
taken to be a prime ; therefore F which is lefs
than it felf A ; contrary to the Hyp.

PROP. XV.

A, 9. B,iz. C, 16. If three numbers continually
D, 3. E,4. proportional 4, B> C7 be the

leaft of all that have the fame
proportion with them $ any two of them added toge-
ther fb all be prime to the third.
a IS* 7* a Take D and E the leaft in the proportion of
& *. 8. A to B ; £ then AmDq, 8c b C=Eq, JandB=DE.

But

1

Digitized by Google

EUCLIDE'* Elements, i8f

fiat becajxfe D c is prime to E, d therefore fhall c 2. 47*
D -f E be prime to both D and E. * therefore d jo. 7*
DxD-t-t e-rz Dq-r DE (f A-hB) is prime to * 26. 7.
E, and fo to C or Eq. Winch was to be dem. e 2.

£ In like manner DE+-Eq (B+C) is prime f before
to D, aad confequeotly to A = Dq. Which was g 17. 7.
*a be dem. n 26. 7-

Laftly, becaufe JJ A is firime to D-+-E, it fhall k 4. i.
alfo be prime to the fquare of it k Dq 2 DE 1 30. 7.

Eq (A -+ 2 B C j) / wherefore the faid B
fhall be prime to A-hB-+C,/ and fo likewifc
to A h- C. Winch was to he dem.

PROP. XVL

A,*. B, $. C — If two numlers Ay B> he

phvie to one another, the fe-
cond B Jball not he to any other C, as the firjl A is
to the Jecond B.

If you affirm A. B R C. then whereas A
and Biare the leaft in their proportion, A ia *J»7*
ihall meafure B as many times as B does C ; b 21 . 7.
but A c meafures itfelf alfo $ therefore A and B c & ax.
are not prime to one another, againft the Hjf.

PROP. XVII.

If there he

A,8. B,i2. C,i8. D,Z7. numbers bow

many foever tn

continual proportion Ay B9 Uy D, and the extremes of
them Ay D he prime one to another y the laJtD Jball
ne the to any other number £, as tbefirjl A if to the
fecond B.

Suppofe A. B D. E. then .alternately A*

D B. E. therefore feeing A and B are the

leaft in their proportion, A h fhall meafure a 23. f*

By c and B likewife C, and C the follow- b 2r. 7.

ing number D, d and fo A fhall meafure c 20 Jef*7*

the faid number & Wherefore A and Dd 11.or.7-

are

Digitized by

j #6 '-The ninth Book of

are not prime to one another, eontrary to the
Hypothefis.

»

PROP. XVIII.

1

A, 4. B, 6. C, 9. Two numbers being given
Bq, 36. Ay B, to confider if there may

be a third number found pro-
portional to them C.
-a 9. ax. 7. If A meafure Bq by any number C, a then
b 10. 7. ACrrBq. from whence > it is manifeft that A*
B :: B. C. Which )vaf to be dem.
A,6. B,4« Bq,i<5. But if A do not meafure Bq,

there will not be any thir4
proportional. For fuppofe A. B :: B. C. a then

Bq

c 7. ax. 7. AC = Ity. * and confequently—^C. namely A
raeafures Bq. Which is againfi the Hypoth.

PRQF, XIX.

A, 8, B,i*. C, 18. D,*7. Three numbers bewg
BC, iltf. £iz/en -5, C, to eon-

fider if a fourth pro^
fortional to them D may be found,
a 9. ax. 7. If A meafures BC by any number D, a then
b ax.19.7. AD BC 5 £ therefore it appears that A, B ::

C. D. which was required.

But if A do not meafure BC, then there can
po fourth proportion^ be found ; which may
be fliewn as in the preceding Prop.

PROP. XX.

1*'

A,2. B,?« C,f. More prime numbers may begiven

D, go. G than any multitude rvhatfoever of

prime numbers J, B ^propounded.
a 7. a Let D be the feaft which A, B, C, meafure ; If
b $j. 7. D-fi be a prime, the cafe is plain; if computed,
b then feme prime number , conceive G, mea«

fures

1

\

Digitized by Google

EUCLIDF* Elements. 187
Cures D -+ 1 5 which is none of the three A, B,
C j For if it be, feeing it c meafures the whole c fuppof.
D-fiji and the part taken away D, e it Hall d eonftr.
alfo meafurethe remaining unit. Which is abfurd.t izasx.7.
Therefore the propounded number of prime num-
Ipers is increafed by D 1 j or at leaft by Q.

PROP. XXL

5 S ? 5 * *
A ••••• E ••••• B-.F...C..G.. D2a

• ♦

If even numbers, how many foever, AB, EC, CD,
be added together , the whole AD Jball he even.

a Take EB = ,l AB, and FC = f BC, and
GD = i CD. * it is plain that EB -+ FC -\- a 6. ief. 7*
GD m ± AD. c therefore AD is an even num- b 12. 7.
ber* Which was to be dcm. c 6. ief. 7.

PROP. XXII.

-

A «,%.«t*«F. B...... G • C.i.. If. D«. L. E iz

9 7 s J

If odd numbers, how many foever, AB, BC9 CD,
D£, be added together, and their multitudes even9
the whole alfo AE JhaU be even.

A unit being taken from each odd number,
tjiere will a remain AF, BG, CH, DL, evert
lumbers, b and thence the number compounded a 7-
of them will be even, add to them the c even b 2I* 9«
number made of the remaining units, and the c ht*
d whole AE will thereby be even. Which was d
to bedem.

1.

PROP.

Digitized by Google

m The ninth Book ef

PROP. XXIII.
7 J i If odd numbers, horn

A»m«.B.....C.»E.Di$. many foever, AB, BC%

} CD, be added' together x

and the multitude of
, them be odd, the whole AD Jball be odd.

For CD one of the odd numbers being tafcui
away, the number compounded of the others AC
a 22, 9. a is evcn< whereto add CD — 1, b the whole
*lj*r ^ 1S cven5 wherefore the unit being re-
e 7. defy, ftored, the whole AD e will be odd. Winch
wm to bcdem.

PROP. XXIV.
4*1 If an even number AB be

A....B,.... D.C10. taken away from an even
6 number AC, that which re-

mains BC Jball be even.
a j.def. 7. For if BD (BC - 1) be odd, a BC (BD -+ 1)
will be even. Winch was to be dem. But if you
b hyp. fayBD is even, becaufe AB b is eves, c thence
c xi. j. AD will be fo ; a and confequently AC (ADt+r)
will be odd, contrary to the Hypoth. therefore £C
is even. Winch was to be dem.

PROP. XXV.
6 1 3 If from an even number

Af~~D.Co.Bio. AB, an odd number AC he
7 taken away, the remaining

number C& fiall be odd.
a 7. def. 7. For AC-i (AD) a is even, b therefore DB
bM-°- is even ; c and confequently CB (DB — 1) ia
c 7. def 7. odd. Which was to be dem. 1

PROP. XXVI.
4 <S 1 If from an odd number

A«a»»C~....D.B ir. AB be taken away an odd

number Q g3 that which
remains AC Jball be even.

For

Digitized by Google

* EUCLIDE'j Elements. t$<?

For AB — i (AD) and CB — • i (Cfy a are a 7. def. 7;
even ; * therefore AD — CD (AC) ts event) 14. 9.

Which was to be done.

■«

PROP. XXVII.

■

146 If from an odd number

A.D....C B if. AB he taken away an even
5 numher CB, the refidue AC

JbaU he odd.

For AB — 1 (DB) a is even, and CB is (up- a 7. defy.
poled to be even ; h therefore the refidue CD is 5 Xa p.
even : c therefore CD -\~ 1 (CA) is odd. Which c j.def.j
was to be dem.

PROP- XXVIII.

A, J* If an odd number A multiplying an

B, 4- even numher B produce a number A£, the

I2. number produced ABJbaU be even.
9 For AB a is compounded of the odd * typ. and

number A taken as many times as a unit is 7*
contained in B an even number, h Therefore Afib Zl- 9*
is an even number.

Schoi

In like manner, if A be an even number, AB
(hall be an even number alfo.

PROP. XXIX.

A, J. If an odd number A multiplying an

B, $k odd numher B7 produce a number AB> the
AB, 1 5. number produced AB (ball he odd.

For AB a is compounded of the odd a I $Jef.J.
number B taken as often as a unit is included in
A likewife an odd number, h Therefore AB isb zj.
an odd number. Which was to he dem.

Sthol

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lyo The ninth Book of

Schol

B, ** /C i. An odd number A meafuring

A, J K 9 * an wen number B9 vie a fur a the

fame by ari even number C.
1 9. ax. 7. For if C be affirmed to be odd, then becaufe a
b 9. B = AC. b therefore B fhall be odd, againjl the
Hypothecs.

B, i$ (c, t z* ^n °^ num^r A meafuring

A, J an °dd number B, meafures the

fame by an odd number C.
a 2.8* 9. For if G belaid to be even, a then AC, or B
will be even, contrary to the Hypotbefis.

B, t6 3. Every number (J arid C) that
A, f ^ 5% meafures an odd number B, is itfelf

an odd number.
For if either A or C be affirmed to be even, B
a 18. 9. a fhall be an even number, againjl the Hypoth

PROP. XXX<

5^ (C,8. (E,4.
A,; v A, j v f*

J/" aw oJi number A meafure an even number Sj
s tijbatt alfo meafure the half of it J).

_s B
* fyf* d Let *t* be =C, b then C is an even number,
b 1. Schoh / A

29, 9. Therefore let E be rr: i C, their Be = CA = z
c 9. a*. 7. EA * = z E). f therefore EA = D$ £ and dpife-

e%# *lemly x=E# m* 1 0 he ieni;

ij.ax.i. PRO P; XXXL

g 7.0*74 A,j. B,8. C,i6. D— If an oddnuntber Abe

prime to any number B, it
fbaU alfo be prime to the double thereof &
- If it be poflible^let tome number D meafure A
a l.fcbcL an(j a t|ien j) ineai'uring the odd number A

9* lhall be odd it felf, b and fo fhall meafure B the
b*0- 9-, half of the even number C. therefc A and B are not
prime one to another. Winch is againft the Hyp- .

Corolh

Y

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EUCLIDE'* Elements.

CoroUm

It follows from hence that an odd number
which is prime to any number of double pro*
greffion, is alfo prime to all the numbers of
that progreflktfu

PROP. XXXII.

191

4 .

t. A, i. B,4. C,8. D, 16. All numbers A \B ,C,

D,&c. in double po-
greffion from the binarie are evenly even only.

It is evident that all theie numbers 1, A,B,Cf
D, a are even, and b 4i , namely in a double pro- & ^ jer ^
portion, cznd fo every lefs meafures the greater 5 zoJe'f.j
by fome one of them. <i Wherefore all are even-c T1] J
ly even. But for that A is a prime number, e j fafcfm .
no number belides thefe fhali meafure any of e j" '
them. Therefore they are evenly even only* * " ^
Which was to be dem.

PROP. XXXIII.

A, jo. B,i $4 If of a number A, the half B U
D — E - - odd, the fame J is evenly odd only.

Being an odd number B a mea- ,
Xures A by two an even number, b therefore B is? ^ \ f *
evenly odd. If you affirm it to be evenly even, 2* ' '
c then fome even number D meafures it by an j * e*'7m
even number E. whence z B d — A, d <=. DE. e ^ ^#
wherefore i. E :: D. B. and therefore as 2 /mea*f
fures the even number Ef g fo D an even num- /}
ber meaftires B an odd. Which is mfoJfibU. 8 *o<**f*7*

PROP. XXXIV.
A, 24. 7f ^I7i even number A be neither doubled
from two, nor have its half {art odd, it u
both evenly even and evenly odd.

It is undoubtable,that A is evenly even,becaufe
the half of it is not odd. But becaufe, if A be
into two equal parts, and its half agaiii

inw

Digitized

t9i Tie nfotk Book of

7,&/#7.into two equal parts and fo on,we lhall at length
light upon feme a odd number (got upon the
number two, becaufe A is not fuppofed to be
doubled upward from two) which lhall meafure
b I fchxQ. A by an even number, fo b other wife A it felf
k 3 fhouldbeodd, againjt the Hypotb. Therefore A
ft alfo evenly odd. WTnch was to be dem.

PROP. XXXV.
«A d«

4 8
( £«••• F •««.•••« G i z.

9 6 4 8
33 H «..«.« L «... K •••••••• N27.

If there be numbers in continual proportion how
inanyfoevet A, BG, C, DNr and the number FG be
taken from tbefecond, and KN from the laft, equal
to the firjl A\ as the excefs of the fecond BF is to
tbefirfi A, fo JbaU the excefs of the laftDKbe to
all the numbers tl>at precede it, -4, BG> C.
From DN take NL = BG, and NM = G.
a brt. Becaufe DN. C (HN) a :: HN. BG (LN) a :r
b 17 <. LN (BG.) A (KN.) b therefore by dividing
ci2 I. each, lhall DH; HN :: HL. LN :: LK. KN. c
d i.'ax.t. wherefore DK C h- BG A:: LK (d BF.) KN
(A.) JKhicb was to be dew.

Coroll,

e 18 ? Hence e by compounding. DN -^-BG *\~ C.
A + BG + C::BG. A.

-

PROP. XXXVI. *
j. A,2. B>4« C,8. D,i&
Sf ji. G,62. H,i24. L,248. F,4*&
M,?i. N,4^-

p..L a--- , . .

If from a unit betaken bow many numbers faever
l,^,B,C,D, in double proportion continually %unul the
whole added together E U aprmemmber yand if this

whole

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EUCLIDE'i Elements.

whole E multiplying tlx laft produce a number Ft that /
which is produced F Jhall be a perfecl number.

Take as many numbers E, G, H, L, likewife
in double proportion continually ; then a of c-a 14. 7.
quality A. D :: E. L. b therefore AL^DE c ~ b 19. 7.

F. d whence L= £ Wherefore E, G, H, L, F, CA J? ,

£ Q J % cix* 7»

are 45 in double proportion. LetG— Ebe=:M,
and F ~ E = N 5 e then M. E ::N. E-+G4-e,( Q
Hh-L. / But M =r E. £ therefore N= E-f , a *
G -hH^L h therefore F = 1 -h B - C - Dff f4 /
-^E-4.G + H-4-L=E^N. Moreover be.^2.^r^
caufe D meafures DE (F) / therefore every £ -*
one, 1, A, B, C, m meafuring D, as m alfo E,j n.^V

G, H, L, does meafure Fi And further,uo otherm ri g# *
number meafures the faid F. For if there do,let n n * -
it be P, which meafures F by Q. n therefore PO" ;1 7#
= F == D. F. 0 therefore E.O :: P.D. therefore ° \9' 7"
feeing A a prime number meafures D, p and fo£ £ ?r
no other P meafures the fame, q consequently^ '

E does not meafure Q. Wherefore E being fup-
pofed a prime number, r it fhall be prime to Q?f \ ' 7"

B, C. Let it be B, feeing then of equality B.
D :: E. H :: * and fo BH = DE = FzrPQ. x* Ip' 7-
and fo alfo Q. B :: H. P. y therefore H — P. y I4* 5"
therefore P is alfo one of them A, B, C, &c.
agamft the Hyfothefis. Wherefore no other be- z xzJef.j
fide the forefaid numbers meafures F, and z con-
sequently F is a perfeft number. Winch was to
he demonftrated.

Tlx End of the ninth Book.

IS THE

t

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?94

THE TENTH BOOK

OF

EUCLIDE'j ELEMENTS.

4 Definitions. '; ' '

L OmraenfuraMe magnitudes arc thofe,
■ ' which are meafured by one and the
\J fame meafure.

Zfa ?jote of compienfuralility is
^ XL 2? ; /for* w, *fo line JofS foot is
commenfurable to the line B of I j /bof a
hecaufe D a line of one foot meafures both
A and B. Alfo V 18 *tl aJ $q ; hecaufe
<J z meafures both V 18 and V For *J

■■5 = ^9=?- *«* V '| = V*S =
5. wherefore y/ 18. V 5^ :: $• 5*
* II. Incommenfurable magnitudes arc
fuch, of which no common meafure can be
found

Incommenfw ability is denoted by this mark l"D-;
as *J 6 *xl 15 *7ja* «r , ^/ 6 is incortimenr
furable to the number 5, or to a magnitude defigncd
by that number , hecaufe there is no common iriea-
fure of them j as fiaU appear hereafter.

Ill* Right lines are commenfurable in power,
when the Tame fpace does meafure their fquares.

The

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EUCLtDE'j Elements.

W The markof this commenfur ability
11 is -tj. ;ar JB iGf. CD. i. e. the
line Jot of 6 foot is in power com-
3> menfurable to the line-CD, which
U&is exprejjed by <J io. becaufe £ the
$ace of one foot fquare docs as well
meafure ABq as the reft angle
XT (20) to which the Jquaie of
the line CD 20) is equal. The
fame note "£J- fometmes Jignifies
commenfur able in power only*

IV. Lines incommenfurable in
power are fuch \ to whofe fqua/es
no fpace can be found to bea common rneafuie.

This incommenfur ability is denoted thus ; s "IJ-
v jsJ 8 i. e. the numbers or lines 5, and v ^ % are in-
commenfurable in power, becaiife their fquares 25
and y/ 8 are incommenfurable.

V. Fiom which it is manifeft, that to any
right line given right lines infinite in multitude
are both commenfurable and incommenfurable 5
fome in length and power, others in power only.
The right line given is called a Rational line.

TJjc note of which is p.

VI. And lines commenfurable to this linef
whether in length and power, or in power only,
are alfo called Rational, p.

VII. But fuch as are incommenfurable to it,
are called Irrational.

And denoted thus p.

VIII. Alfo the fquare which is made of the
faid given right line is called Rational, py.

IX. And likewife fuch figures as are commen-
furable to it, are Rational, p*.

X. But fuch as are incommenfurable, Irratio-
nal, pot,.

XL And nhofe right lines alfo, which con*
tain them in power, are Irrational f> .

If * Stbot.

19 f

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i

196 The tenth hook of

SchoL

Tim the loft fcven
definitions may T>e ren-
dred more clear by an
example, let there be a
circle ADBP, wUfe ft~
midiameter is CB,, T9r
fcribe therein the fides of
the ordinate, figures, as
p of a Hexagone BP. of a

triangle AP, of a fyuare BD, of a Pentagone FD.
Therefore, if according to the 5. def. the femidiame-
ter CB he tlx Rational line given, expreffed by the
number i. to which the other lines BP, AP,BD%
2ccr.iK./L.F^ are 10 *■ comia™d, then BP a~BC~zm
b 47 r vlmefoYt BP is p T3 BC, according to the 6. def.
*' ' Alfo JP b — V » {for ABa (16) - BPq (4) =
iz) therefore AB p 'tj. BC. likewife acco* ding to the
6. def. and APq (12) is aV by the 9. def. Moreover
BDb - v DCq BCq -v'8; whence BD is
p ^BC i andBDa pr. L*//y, FD? - f«>
f o. {as Jball appear by the praxis to be delivered
at the 10. 1 Jball be fV, according to the 10. def.
and confequcmly FD ass y : 10 — ^/ zois p, ac-
cording to the ii. def.

•

.1 • > '

. V

A Pojlulate.

That any magnitude may be fo often multi-
plied, till it exceed any magnitude vrhatfoever
of the -lame kiad.

• - «

Axioms.

T. A magnitude meafuring how mm y mag-
nitudes foevcr, does«a)fg meafurc tfeat which
is comuofed of them.

2. A

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EUCLIDE'j Elements.

2. A magnitude meafuring any magnitude
whatsoever, does likewite meafure every magni-
tude which that meafures.

3. A magnitude meafuring a whole magnitude
and a part of it taken away, does alfo meafure
the refidue.

*

prop. r.

w Two unequal magnitudes AS, C, heing
t &ven> if from the greater AB there be taken
.iway more than half ( AH) and from the
\JQ ' yefidue (HB) be again taken away more
than half (Hi) and this be done continu-
\K atty% there Jball at length be left a certain
IF 1 magnitude IB, lefs than the lefs of the
I magnitudes jir ft given C.

a Take C fo often, till its multiple a
ACS} DEdo fomewhat exceed AB, and there
be DE-rFG— GE-C. Take from AB
more than half HA,and from the remaindei HB
more than half HI, and fo continually, till the
parts AH, HI, IB, be equal in multitude to the
parts DF, FG, GE. Now it is plain, that FE,
which is not lefs than I DE, is greater than HB,
which is lefs than <l AB ~3 DE. And in like
manner GE, which is not lei's than ^ FE, is
greater than IB -3 ; HB. therefore C/orGEr-
IB. Winch was to be dem.

The fame may alfo be demonftrated, if from
AB the half AH be taken away, and again from
the refidue HB the half HI, and fo forward.

N j PROP.

i?8

PROP. II.

a T. TO.

bhyf.

*r*D Two unequal magnitudes being given (JB9

~ CD) if the lefs AB^ be continually taken front
^ .."F the greater CD, by an interchangeable fuh-
i^.. JtmXjoTi, and the refidue do not meafure the
magnitude going before } then are the mag*>
niiudes given incommenfurable.

If it be poffible, let fome magnitude
E be the common meafure. Thenbecaufe
AB taken from CD, as often ?s it can be,
'leaves a magnitude FD lefs than it felf,
AC EandFD taken from AB leaves GB, and fo
forward ; a therefore at length fome magnitude
GB~Efhall beleft. therefore Ei meafuringAB,
C z.ax.io.c and fo CF, b and the whole CD, d {hall alfo
meafure the refidue FD. c confequently alio AG;
d 3.^.10.^ wherefore it fhall likewife meafure the remain-
der GB, lefs than it felf. Which is abfwa\

PROP. III.
Two comvienfurahle magnitudes being
given JBy CD, to find out their greatejt
common meafure FB.
EBr Tak( AB from CD, and the refidue
T . LD from AB, and FB from ED, till
FB meafure ED (which will come to
pafsat length, a becaufe by the Hyp.
. AB tl CD) FB ftall be the magni-
tude required.

For FB b meafures ED, e and fo alfo
AF 5 but it meafures it felf too, ^there-
fore likewife AB, c and confequently
CE, d and fo the whole CD. Wherefore FB is
the common meafure of AB, CD. if you affirm
Q to be a common meafure greater than that,
e2.tf*.rc.then G mealming AB and CD, e meafures alfo
t ^sr.tc. CE and /the remainder ED, e and fo AF ; and/
confequently the remainder FB, the greater the
lefs, Which is abfurd*

\

a z. ie*

b covjtr.
C 2. aec.io.

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EUCUDFi Elements] i99

CorolL y .

Hence, A magnitude that meafures two mag-
nitudes, does alio meafure their greateft common
meafure,

* •

PROP. IV.

Three commen fur able viagnitudes being given J>B9
C, to find out their greateft common meafure.

a Find out D the greateft common meafure a 3. 10.
of any jwo A, B ; a alio E the greateft common
meafure of D and C. therefore E is the magni-
tude fought for.

a For it is clear, E meafuring D and C, b does bcon]tr.&
meafure the three A,B, C. Conceive another 2.. ax* 10.
magnitude F greater than that to meafure them ;
c then F mealures D, c and consequently E the c cor.j.io.
greateft common meafure of D, and C, the
greater the lefs. Which is abfurd.

Coroll.

Hence alfo it appears, that if a magnitude
meafure three magnitudes, it lhali likewife mea-
fure their greateft common meafure.

■

PROP. V.

A — - — — — — D. 4 . Commenfurable mag-

C» F.i. nitudes J, B, have

E.g. fuch proportion one

to another, as number hath to numbeu

a C beiug found the greateft common meafure a 3. 10.
of A,B ; as often as C is contained in A and B,fo
often is 1 contained in the numbers Dand E *, b bij.def.j.
therefore C.A 51.D5 wherefore inverfely A.C :: '
D. r. b but likewife C. B :: I.E. c therefore of e-
quality A. B :: D.E N- W. IK to be dem.

N4 PROP.

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2oq The tenth Book of

; , PROP. VL

E F,i. // two magnitudes

A ■ » » C,4^ jf, foue pro-

B D,$. portion one to another

as number C has to
numberDjhofe magnitudes J yBJb all becommenfurable.
nfchio.6. What part i is of the number C, a that let E
b conftr. be of A. Therefore becaufe E.A b :: i .C. and A.B
C hyp, c :: C.D. d therefore of equality (hall E. B :: i. D.
d 21. J. Wherefore feeing i e meafures the number D, f
e <>.ax. 7. likewife E mealures B} but it£ alfo meafures A*
f zoJeF.y. h therefore A XL B. Winch was to be dem.
g conftr.

hiJefao. PROP. VIL

A ■ Incommensurable magni-

B tudes A, B9 have not that

proportion one to another ,
which number has ' to number.

a 6. 10. If y°u aflfirm A. B :: N. N. a then A TL B

againjt the Hypothecs. ^ '

>

PROP. VIII.

A If two magnitudes JyS, have

B — not that proportion one to ano-

ther, which number has to num-
bo\ thofc magnitudes are incommenfurable .
a >. jo. Conceive A *~o_ Ba then A. B :: N. N. con-
trary to the Hypothefis.

PROP, IX.

A ■ The fquares defcribcd of right

£ lines cQvimenfurable ill lengthy

E* 4. have that proportion one to ano-

P? ther% that a fauare number has

t to a Jrj uatt number. Am Ifpiares , which have that pro-
portion

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EUCLIDE'* Elements: aoi

portion one to another, that a fquare number has^o

a fquare number, Jba.il alfo have their fides commen-

furable in length. But fuchfquares a* are made of

right lines incontmenfurable in length, have not that

proportion one to another, which a fquare number

as to a fquare number. And fqxuxrts which have

not fuch proportion one to anothex as a fquare num-

lor has to a fquare number, hate, not their fides

commenfurable m length.

I. Hyp. A. ul B. I fay Aq. Bq Q. Q.

For a let A.B:: number E. number F. therefore

Aq A . N E . j Eq , ^
twice) J? twice, d- e there- b zc. 6.

foreAq,Bq::Eq.Fq::Q.CL. Winch was to be dem. 5Af*J*

z.Hjtu Aq.Bq::Eq.Fq::Q:Q. Ifay A"LLB.For ^

A . rAq, Eq, E . . - *TI' h
^ twice g b r=i - twice, i there- f 20. 6.

jo ' r>q rq r c ;;y*

fore A. B :: E. F :: N. N. k wherefore A'uB| g.
Which was to be dem. . i fch.zi ?

3. ATLC. I deny thfct Aq. Bq;: Q. Ctj/g, ic!
For fuppofe Aq. Bq Q. Q. then A 4-Q- B, as

is fhewn before, againfi the Hyp.

4. Hyp. Not Aq. Bq :: Q. Q. I fay that A*n-
B, For conceive A ~o B. then Aq. Bq Q. Q?
as above, againfi the Hyp.

LoroU,

Lines XL are alfo but not on the con-
trary. And lines "TO- are not therefore "5-. but
nrj- are alfo Tx.

PROP.

1

T

•4 t 1 I • ..

U /

Digitized by Google

The] tenth Book of

I

a $. 10.
b6. io.
c 7.10.
d 8. 10.

PROP. X.

If four magnitudes be proportional (C,
J :: B. D) and the firfi C be commcnfu-
rable to the fecond Ay the third B JbaU be
commen fur able to the fourth D. And if
1 the firfi C be incommensurable to the fc~
1 \ cond J9 alfo the third B JbaU be incom-
C A B Dmenfurable to the fourth D.

IfC XL A, a thenC.A N.N b:: B.D. lb
therefore BuD. But if XL A, c then fhall
not C. A . : N. N :: B. D. d wherefore B
Which was to be dem.

Lemma 1.

To find out two flane numbers, not having the pro-
portion which a fquare number hath to s fquare.

Any two plane numbers not like will fatisfy
this Lemma, as thofe numbers which have fuper-
particular, fuperbipartient, or double propor-
tion i or any two prime numbers. See fch.zj. 8.

B, S.

C, j.

K
H

Lemma
-1
1

2.

1
1

1

R

M

To find out a line HR> to which a right lincgiven

KMhaththeproportienof two numbers given B,C.
a/c/;.ro.6. a Divide JCM into as many equal parts as

there are units in the number B. and Jet as
b 5. i. many of thefe, as there are units in the number

C, b make the right line HR, it is manifeit

that KM. HR B. C.

Lemma

To find out a line D, to the fquare of which, the

2}iJem.lo.fV'a}e °fa Sjven %™ hath the proportion

" " of two numbers given B , C.
b 1, 6. Allow B. C a ;: KM. HR. and between KM
c 10 6 anc* tlR» * * mean proportional D. Thefre-
J Jj£ fore K Mq. Dq c :: KM, HR d B. C4

P R O P«

1

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« . *

EUCUDEV EUmtnti. §o|

PROP. XL

B, 2o. To find two right line*

C, i<. incommenfurahle to a tight
line givpn Ay one D in
length only y the other E in

power alfo.

1. Take the numbers B, C, a to that thereai.fcw.ro.
be not B. C .*: Q Q. * and let B. C Aq. Dq. cto.

it is plain that A no. D. But Aq d TL Dq. b^lemat.
Which was to he done* 1 o.

2. <*MakeA.E.:E.D. I fay AcrtLEq, ForA.c 9. i<x
D::Aq.Eq. therefore fince A-p.D, as before : fd tf.iou
therefore Aq'TxEq* Which was to he done. d 1$. tf.

e zo. <$,

PROP. XIL fio.10,

I Magnitudes (J> B) commenfurable to the
fame magnitude Cf are alfo commenfuralle
one to the other.
Becaufe A tl C, and C*ru B, a let A. a j. ia
D,r8. E,8. C.v N.N .v D. E, and C.
. j ,F,2. G,j. B::N. N.vF. G. itakeb^fc
I I I H,$.I,4»Kj6. three numbers H. I, K, the
A B Cleaft H- in the proportions of D toE,
and F to G. Now becaute A.C c ;: D.E c :: H.I,c confir.
and C. B c :: T* G I. K. d therefore of equality d 22. 5.
A.B :: H.K N. Ni e therefore A no. B. Which c 6. icx
was to be dem.

• Schol

Hence, Every right line commenfurable to a
rational line is alto it f elf rational. And all iz.ic.and
right lines rational are commenfurable to one def tf,
another, at leaft in power. Alfo, every fpace
commenfurable to a rational fpace is rational
too : and all rational fpaces are commenfurable def. 9.
one to another. But magnitudes whereof one
is rational, the other irrational, are inepmmen- def 7. ^
furable amongft themfelves, ic.

1

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The tenth Bool of

PROP. XIII.
A ■ — Jf there he two magnitudes A,

C t » By and one of them A commevfu-
B 1 1 * ■ rable to a third C, but the other

B incommenfurable, thofe mag-
. nitudes A, B are incommenfurable.
J tyf* Conceive B TL A. then being C a XL A. b
1 I*, io. therefore C uB, againjt the Hyp.

PROP. XIV.

I

If there he two magnitudes commenfurable
A,B i and one of them A incommenfurable to
I any other magnitude C, the other alfo B Jball
a ^* \ I r incommensurable to the fame C.
b 12. io. j Imagine B t^-C. then for that \a ~zx
I * 1 B, b therefore AtlC, againjt the Hyp.
ABC

PROP. XV.

A ■" ■ - mm* If four right lines be pro-

J* portional (A. B :: C. D.) and

^' the fir ft A be in tower more

*2 ■ » than the fecond B by the f quart

of a right line commenfurable
io it felf in length, then alfo the third C Jball be
more in powei than the fourth D by thefauare of a
\ line commenfurable to it felf in length. But if

tFefirft A be more in power than the fecond B by the
fquare of a light line incommenfurable to it felf in
length, then Jball the third C be more m power than
the fourth I) by the fquare of a tight line incommen-
lurable to it felf in length,
zhyp. For becaufe A.B a :: C. D. b therefore Aq. Bq ::
b 22. 6. Ct.Dq. c ther fore by divifion Aq Bq. Bq :: Cq
C17. 5^ — Dq.Dq. d wherefore *J : Aq— Bq.B.-.V: Cq —
d 22. 6. D ; D.e and lo inverfely B.^: Aq Bq:; Dy:Cq.
e cor. 4.^. Dq. / therefore of equality A. 4 : Aq Bq C.
f 22. 5. V: Cq - Dq. confequently if A TX , or x^. V Aq,

Bq,

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EUCLIDE'* EUmentf. 20$;

— Bq. g then likewlfe C TXf or TX <J : Cq — o 10. tow

Oq. Winch was to be dem. p

PROP. XVI.

1 — C If two magnitudes

B commenfurable
D— — jfc* cbmpofed, the whole

magnitude AC Jball be
commenjurable to each of the farts AB, BC. And if
the whole magjiitude AC be commenfurable to either
of the farts AB, or BC, thofe two viagnitudes given
at Jirjl AB, BC, Jball be commenfurable.

r. Hyp a Let D be the common meafure of a J. 10J

AB, EC ; b lb D rneafures AC. and therefore b \.ax.\o.
AC TL AB, and BC. Which was to be dem. c i.<fc/.ic%

2. Hyp. a Let D be the common meafure ofd j.d^.io*

AC, AB. d therefore D meafures AC AB
(BC) and confequently AB -el BC. Winch was
to be dem.

Coroll.

Hence it follows, if a whole magnitude com-
pofedof two, be commenfurable to any one of
them, the fame fhall be commenfurable to the
other aJfo.

nop. xvii.

A j C If two ineommenfurtbb

IB "~ viagnitudes AB, BC, be com-

H •" t°fzd, the whole magnitude

alfo AC fiall be incommeu-
Jurable to either of the two parts AB, BC. And if
the whole magnitude AC be incommen fur able to one
of them AB, the magnitudes firjt given ABy BC%
pall be incommenfurabtc.

1. Hyp. If it can be, let D be the common

meafure of AC, AB. a therefore D meafures AC* .1

-AB (BC) b and therefore alfo ABtx BC,J'fg £
agamjt the Hypoth, *pt*f*+

Z.Hyp.

.r; .'
»

Digitized by

2o6 The tenth Book of

c i<5. io. HyP* Conceive AB tj. BC. c therefore AC

TX AB, againjl the Hyp. , .

CoroU. ^

Hence alfo, If one magnitude, compofed of
two, be incom men fur able to any one of them,
the fame alio dull b» incommenfurable to the
other.

PROP. XVHt

If there be two
unequal right lines
AB, GK and upon
the greater AB a
parallelogram ADB
equal to the fourth
fart of a fquare
G xj v made of the lefs line

r GK^ andwantingin
figure by a fquare,be applied, and divide the [aid AB
into parts commenfurable in length AD, DB -y then
fhall the greater line AB be more in power than the
lefs GK by the fquare of a right line FD commenfu-
rable in length to the greater. And if the greater
AB be m power more than the lefs GK by the fquare
of the right line FD commenfurable unto it Jeif in
length j and a parallelogram ADB equal to the fourth
part of the fquare made of the lefs line GK, and
wanting m figure by a Jquare, be applied to the
\ greater AB, then fhall a divide the fame into parts
& 10. 1. AD, DB commenfui able m length.
i 28.6. a Divide GK equally in H, and b make the
c8.2. reftangleADB GHq. Cut off A F .DB. then is
&<mjtr.& ABq c = 4 ADtf (4 GHq or GKq) -FD Now
4. z. inthefiritpi.ice, ir AD m. DB. men lhall AB e
ci&io. -ixBD<: £ 1 DB/(AF DB, or AB— FD) b
g cor. if. thereto. e AB "UL FD. Winch was to be dem. But
io. feconoiy, if AB v. FD, b then (hall AB ' + AB
k 1^10. FD (2 DB) k theirioic AB DB. / where-
1 16. io. fore AD "DlDB. Which was to be dem.

PROP*

1

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EUCUDE'j Elements:

207

PROP. XIX.

If there le two
right lints unequal
AB, GKy and to the
greater AB be appli-
ed a parallelogram
ADB equal to tl*
fourth fart of &
fquare made upon
the lefs GK, and
wanting in figure by a fquare, and alfo thus applied
divide the j aid AB into parti AD, Do incommenfw*
table in length ; the greater line AB foall be in
power more than the lefs GK by the fquare of the
right line FD incommen fur able to the greater in
lengthXAnd if the greater line AB be more in power
than the lefs GK by the fquare of a right line FD
incommensurable unto it felf in lengthy and if alfo
upon the greater AB be applied a parallelogram
ADB eaual to the fourth part of the fquare of the
lefs GKand wanting in figure by a fquare, then
fhall it divide the faid greater line AJS into parts
mcommenfurable in length AD, Q/?.

Suppole all the lame that was done and faid
in the prec. Prop. Therefore firft, If AD Tl
DB, a then fhall AB Tl- DB. b Wherefore AB a 11. 10.
Tl z DB (AB - FD) c therefore AB Tjl FD. b 13, iq.
Winch was to be dem. c cor

Secondly, If AB Tl. FD, then AB TL AB— ro.
I'D (iDBO <* wherefore ABtlDB, sandcon- d 13. 10.
tequently AD TL DB. Winch was to be dem. e 17, ft*

1

E ROP. , .

«

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tog Zfc tmtb Book of

PROP. XX.

A A re8 angle BD compre-

TJ hended under right lines BCi

CD, rational and commenfu-
rable in length, according to
one of the forefaid ways, is
rational.

■W •! fi. Let A be given p, and

a 46. 1. * the fquare BE de^ribed upon BC. Becaufe DC.
b 1.6. CE (BC) * :: BD. BE. and DC c a BC, d
c therefore fhall the reftangle BD be tl fquare

d 10. 10. Bfc. wherefore feeing the fquare BE e ~cl Aq,
c hyp. awrffliall alfo BD be no. Aq. and £□ the reftangle
9 10. BD is pr. #7;/V/; war to be dem.
f 12. 12. Note, There are three kinds of lines rational
commenfurable one to another. For either , of two
lines rational commenfurable in length one to the
other, one is equal to the rational line propounded, or
neither of them is equal to it, notwithftand'ing both
of them are commenfurable to it in length \ or laftly,
loth of them are commenfurable to the rational line
given only in power. And theft are the ways which
the prefent Theorem fpeahs of.

In numbers, Ut there be BC V 8 (2 J 2) and
CD V 18 ($ V 2) then fliallthe reftangle BDr=
V 144 SS 12.

PROP. XXL

3f If a rational rectangle

DB be applied to a ra-
tional line DC, i$ makes
the breadth thereof CB

X 1 - L J -4 rational, andcommenfu-

ai.6. *^ 0 » <* raifc in iength to that

b hyp. line DC, whereto DB is applied,
cfch. 12. Let G be propounded f, and the fquare DA
10. defcrlbed on BC. becaufe BD. DA :: a BC. CA ;
d 10. 10. and BD, DA b are pet c and fo TL. d therefore
efcb. 12. BC no. CA. but CD (CA) is p. e therefore BC
10. is L Wlnth was to be dem.

In

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EUCLIDE'f Elements.

Id numbers, let there be re&angle DB. 12*
and DC, V 8. then (hall CB, V 18. but V 18

Lemma*

A — 2b j(M owf two fight lines ratio*

B . ■ ■ ■ wflZ commenfurahle only in power.

C Let A be propounded p. <f a ft. toi

Take B A, * and C ~g- R i it if clear chat bfcb. i%4
B and C are the lines required. ie«

PROP. XXtL ;

A reSangle DB comprc~

irrational : and the right
line H, which containeth that reftangle in f owct
is irrational, and called a Medial line.

Let Q be the propounded />, and the fquare
DA defcribed on DC> and let Hq— DB. Becaufe
AC CB a :: DA, DB. b and AC H. CB, c fhall t I. 6*
be DA tx DB (Hq,) d but Gq tl DA. e b hyf.
therefore Hq *TX Gq. / wherefore H is p. Which c ip. IO;
w<v *o to. and let it be called a Medial line, d hyp. and
becaufe AG. H :: H. CB- 9 jufc/Iio*

In numbers, let there be DC, 3. and CB, ^/6i e ij. 10.
then lhall the reftangle be DB (Hq)y $4« f
wherefore H is v *J 54.

The note of a medial line is of a medial
jreftangle p? , of more together

Every re&angle that can be contained undet
two right lines rational commehfurable only in
power, is medial, alt ho* it be contained under two
right lines irrational : and every medial reftangk
m^y be contained under two right lines rational,
tommenlurable only in power, as for example,

O , th*

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2IO

r T .T*

a fch. ii.
10.

b t« i.
c 14,6.
d n. 6.

f fch. 11.
10.

g 10. 10.
n 1 1.
10.

k 1. tf.

1 10. 10.

10.

n 15. 10.
o i. 6.

p lu. 10.

* 7&e f e nt h Book of

the aJ 24 is becaufe it is contained under y/
3, and 3, which are p, "t^-. altho' it may be
contained under v v 6\ and t> ^ 96 irrationals ^
fcr ^ 24 ^uy' $j6=vi/6xv y 96.

PROP. XXIII.

// the reft angle BD
made of a medial line
J, be applied on a ra-
tional line BCft makes
the breadth CD ratio-
nal^and incommen fina-
ble in length to the line
£C, whereunto the rettangle BD is applied.

Becaufe A is a therefore fhall Aq be equal
to fome reftangle (EG) contained under EF and
FG p b therefore BD - EG. c whence BC.
EF:: FG. CD. d therefore BCq.EFq:: FGq.CDq.
But BCq and EFq e are p*, / and fo nc . g there-
fore FGq CDq. Wherefore being FG is p,
7; therefore CD (hall be p. Moreover, becaufe
EF. FG lb :: EFq.EG (BDO for that EF *n. FG,
e (hall EFq be m BD. But EFq m n CDq. n
therefore the reftangle BD "O. CDq. Whence
being CDq. BD 0 :: CD. BC f fhall CD be ^xl
BC. therefore, &c.

PROP. XXIV.

a 1 1. 6.

b hyp. .

ci?. 10.
«1 I. 6.
« hyp.

AriglA line B coik*
menfurable to a me-
diallme A is alfo ct
medial line.

Upon CD p s
make the reftangle
& & E CErr Aq> 0 and the

redlangle CFrrrBq. Becaufe Aq (CE)is^,£and
CD p. e therefore fhallthe latitude DE bep Tj-
CD. But for that CE.CF d :: ED.DF. and CE t tl

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EUCLIDE\* Elements. 21 1

CF, / therefore ED n t)F. g therefore DF isf 10. 1**
f CD, h wheoce the. iedtangle CF (Bq) isg it. and
and fo B is Wind) was to be dem. t 3. l€*

Obf. Tij/ wofe ~Q for the moji partjignifiesh 12. IOi
eommenfttrable in power oniy% as in this and the pre-
cedent demonjtrations, &c.

Hereby it is rtianifeft that a fpace comitienfu*
rable to a medial Ipace, is alfo medial.

Lemma.

A *■ — ■ To find out two right tihei medidt
B ...... Ay By commensurable in length^and

C — — — aljfotwoy Ay C\ commenfuraiie on-
ly in power.

<z Let A be any b take B tl A, and e C * km. iii
-Tj- A. d it appears to be done. ' 10. and

PROP. XXV. hilem.xSn

10.'

JreflangleDB contimed un teM.tb*
der DCy CB iHedial right lines 16* /
commenfurable in iengU), is me~ dconjtr.ty
dial. Z4.;io*

Upon DC defcribe the fquare
DA. Being AC (DC.) CB a .v i t< 5*
DA. and DC TL CB ^ b fhall D A TL DB, b ia fa
t therefc^ DB is ^ Whisk was ta he dem, c 14- ***

i -

- . 1 • * *

t % ■ • ■ • * •

..... . ,

■ * • • •

Digitized by

2IZ

• - < • <

" 7i« f«tfJ& Booi 0/

#

PROP. XXVI.

I

D 1

KM

a 46. 1

A reSangle AC comprehended under medial rig//
lines AKy BC commensurable only in power, is eithcr
rational or medial.

Upon the lines AB, BC, a defcribethefquares
AD,CE; andupen FG i make the reftanglesFH,
bcor.i6.6.~ AD, b and IKzrAC.£ and LM=CE.

The fquares AD, CE, that is, the reftangles
c hyp. &FH, LM, c are yut and TJ,. therefore GH,KM,
24. 10. having the fame proportion d arep, e and tl.
d 23. 10. / therefore GH x XM is jr. But becaufe AD,
e 10. 10. AC,CE. that is, FH, IK, LM, £are 4? 5 /; and
f 20.10. fo GH, HK. KM alfo-f? } k thence HKq = GH
gfch.zz.6.x KM. / therefore HK is p or tl, or TJ- IH
h 1. 6. (GF;) if tl, w then the reftangle IK or AC is
k 17.fi. py. but if tl, n then AC is ^. Which was to
1 12. 10. be dm.

m 20. 10. ' Lemma.
1 22. 10.

A E

If A and Eh

a hyp. andEq, Aq+Eq, Aq--Eq a tl . Andfecondly Aq,Eq*
to. 10. Aq+Eq. Aq— Eq "n. AE and 2 E. For A-Ei::
b 1. 6. Aq. AE b :: AE. Eq. therefore feeing A c *n-E,
c hyp. d fhall Aq "n. AE, e and 2 AE. alio Eq d
?d to. 10. AE, e and 2 AE. wherefore becaufe Aq-+Eq
♦ 14. 10. Aq and Eq ; and Aq — Eq tl Aq and Eq.
1 14. 10. therefore fhall Aq ^ Eq. / and Aq — Eq be
AE, and 2 AE.

EUCLIDE'x Elements.

Hence alfo thirdly, Aq.Eq, Aq -+ Eq, Aq— Eq,
z AE g *tl Aq h- Eq +■ z AE ; and Aq -h Eq g 14. i&.

— 2, AE# and Aq +- Eq •+ 2 AE 'tx Aq -h Eq & 17. loj

— z AE, A (Q; A — E.)

213

XXVII.

^ w&Jitf/ reSangle
AB exceedeth not a me-
dial reftangle AC by x
rational reft angle BB.

Upon EF p, a makea Cor.i6.6.
EG = AB, a and EH
= AC. The re&angles
AB,AC, u e. EG,EH, bb hyp.
mzua\c therefore FG c 2 3. iq,
and FH arepe ^Tf-EF. Whence, if KG ,^2. e.DBd i.ax.u
be iy> e then {hall HG be "o_ HK ^ / wherefore e zr. io»
HG ^ FH. £ and confequently FGq Tl FHq.f ij. 10.
But FH is p. A therefore is FG p. but FG wasg Urn. 16.
p before. Which is contiadiftory. 10.

bfcktz.

Schol. 10'

1

AC B

L

•

1. ^ rational reft angle AE
exceeds a rational reSangle AD
£y <x rational reft angle CE.

For AE TX AD. J there- a Ay/>.
fore AE tl CE. c wherefore b cor. 16.
CE is *f¥. Which, &c. 10.

z. -/ rational reft angle A Dc /<;&. 12
joyned with a rational reSangle iq.
CF ?wdrfeex a rational reftangle
AF.

Da c

and fo AF is py9 Which was to be dem.

fore AF tx AD and CF, c i0,

b 16 .1

c yti .i

O? PROP. 10.

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a i4

tenth Book of

i

t few. zi.

b i$. 6.
<? 12. 6.

d iz, io. A C B

PROP. XXVIII.

To find out viedial lines (C and DJt
which contain a ratiojial rectangle CD.

a Take A andB ^ ncr* * make A.C:;
C.B. c and A.Bv; C. D.* I fay the thing
required is done. For AB (Cq)d is
m d whence C is ,u. but being that A. Be
:: C,D. f therefore C ~g~ D. ^andconr
Dfequentl) D is p. Moreover by inver-
iion A.C B.D. ?. e. C.B B.D. Z> there-

* io. io. fore Bq^CD. But Bq is fo. b therefore CD is^r?
~ x4* io. Which was to be done.

17. 6. In numbers, let A be z \ and B J 6. there-
t Jfq hf 12, fore C is v *J 12. make V 2. V 6 « i V D,
Ot ' or v V 4. *V?<5 v *J 12.D. then fhall D be x> V
ic8. butt; ^12 x vv^io8=:x; ViJ96= V?<S
±5. 6. therefore CD is 6. likewife C. D 1. V
3. wherefore C -5- D.

4 m

I

10.

C 12,(5.

dl7.(5f
e 22, 20.

PROP. XXIX.
To find out medial right lines cotq-
menfurable in power only. D and £yco7^
taining a medial reclangle D£.

a Take A, B, C, i make A.D
b :: D. B. c and B. C ;? D. I fay
the thing defired is performed.
For AB d m Dq. and AB e is pv%
ADBC Etherefore D is p 5 and B / ~cj- C,
£ whence DUE. therefore ft E

Pi

It:

lili

i qonJir.& » Moreover B, C / .v Ml and by inverfion
g 10. 19. D.v C E. i. e. D. A .*: C. £. / therefore
n 24. 10. AC. But AC m js i*y, therefore D£ isp
Yconfir.& yas to be done.

mr4f 5
J i<5, 6,

In numbers, let A be 2a and B,, V ^oo,an(
V' 80. Therefore D is V V Saooo ; and E v *J
12800. Therefore DE =c V V io24o#pouO 35
3200a and R.E V *?? 2, vrherefpre D ng. E.

EUCLIDF* Elements. aif

*i * * . *

A,6. C,rz. To find out / lane numbers, like
£,4. D,8. or unlike.

proportional A. B C. D. it

A, 6. C,5* is.jnanifcft that AB and CD

B, 4» D,8. are like plane numbers. And
AB, 14. CD,4o. you may find out as many un-

• like plane numbers, as you

pleafe, fry help of SeboL 27. 8.

* #• • • • • ,

/ 1

^^^^

§ ' l M'

/ •

JP C "D B

' To find out two fquare numbers (DEq and CDa)
fo that the numier cpmpofed of them (CEq) U

fquare alfo. t

Take AD, DB like plane numbers (of which
let both be equal, or both odd) viz. AD, 14. .
and DB, 6. The total of thefc (AB) fc$Oj the
difference (FD) 18 halt of which (CD) is 9.
a Now the like plane numbers AD, DB, have a 18. 8.
one mean number proportional, namely DE.
therefore it is evident that every of thole num-
bers CE, CD, DE, are rational, and by confe-
quence CEq Q> CDq -\- DEq) is the fquare b 47, 1,
number required. .

Whereby it will be eafy to find out two Iquare
numbers, the excefs of which is a fquare or not
a fquare number, namely by the fame cgnftru-
<ftion c fhall CEq — CDq be = DEq. c }<ax. f

But if AD, DB be plane numbers unlike, the
; • • O 4 medial

Zl6 ' The tenth Book of

medial proportional line (DE) Hull not be a
rational number, and fo neither fhall the excefs
(DEq) of the fquare numbers, CEq, CDq, be a
fquare number,

■

i

Lemma x.

1. To find out two fueb touare numhers 5, C, as
the number compounded of them D is net fquare*
Jlfo to divide a fquare number A into two numhers
$f C, not fquares.

A, j. B, 9. C, 36. I),4f.

i. Take any fquare number B, and let C
be •= 4 B, and DrB^-C, I fay the matter is
done.

For B is Q. by the con ft r. likewife becaufe)
* 24. 8* B.C .v 1. 4 Q. Q. therefore C alfo (hall be 4

fquare number. But becaufe 3 C. (D) C
b w.24.8T j. 4 not Q, Q. ff therefore lhall not D he *
fquare number. Which was to be done.

A, 56. B,24. C, 11. p, F, r.

z. Let A be fome fquare number. Take D,E,
F, plane numbers alike, and let D be = E ■+ F.
make D. E A. B. and.D. F A. C. I fay the
thing 1 emu red is done.

For becaufe D.E-+-F :: A. B ~\- C. aad
a 14. 5. E •+ F. a therefore (hall B C. Now fup-
bziJef.j. Pofe B to be fquare, b then A and B, c and con-
Ci6y& (equently D and E are like plane numbers.
Which is contrary to the Hyp.

The fame abfurdity will follow if C be fuppen
fed a fquare number. Therefore, &V.

ROP,

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EUCLIDEV Elements. ziy

PROP. XXX.

To find out two fuch ratio-
nal right lines AB, AF, com-
menfurable only in power, as
the greater An JbalL be in
power more than the lefs AF
by the fquare of a right line
C ...f E ..... D BF commenfurable in length

to the greater.

Let AB be the line given p. a Take the fquare a r# \em^
numbers CD, CE, fo that CD-CE (ED) be notZp. I0.
Q. b and let there be CD. ED ABq. AFq. In b 3. Um.
a circle defcribed upon the diameter AB c draw r0. i0.
AF, and alfo BF. Then I fay AB, AF, are the c f # 4.
lines required. d covjh*

For ABq. AFq d.vCD.ED, e therefore ABqXL e 6. 1©.
AFq. but AB is p. / therefore AF is alio p. But f jcjJt I2#
becaufeCDisQ: and ED not Q: g therefore fhall I0#
A B be ^ll A F. Moreover by real on of the /; right g 9 1 c
angle AFB " * *~ 1 r

feeing AB<
portion fha

fore AB "tx BF# " Which was to be done.

In numbers, let there be AB,6; CD, 9; CE,4;
wherefore ED,$. Make 9. $ :: $6. (Q:6) AFq.then
AFqAall be 20. and confequently AF V ^there-
fore BFq^ 16 - 20= 1 6. wherefore BF is 4.

PROP. XXXI.

Tofindout typo rationallines
AB yAF commenfurable only in
power , fo that the greater AB
fhall be in power more than the
■j lefs AF by the fquare of a

C E D right line BF incommen fur able

h\ length to the greater.
Let AB be the line given p\ a Take the fquare a 2. lem.
numbers CE, ED, fo that CD r= CE + ED be 19. 10.
not Q. and! in the reft follow the conftruclion of
the pieced. Prop. I fay then the thing required
is done.

For

Af. iVlUICOVCl Uj ICalUJJ Ui II1C IJ ll^lll or O. IQ.

B, is ABq k -= AFa -\- BFq ; therefore f| j" x ^
>q.AFq.v CD.ED. by converfion of pro- ^ 4t t
iall ABq.BFq .v CD.Cfc :; Q. Q. / there- j 2' |0"

Digitized by Google

^jg Tie tenth Book of ,

b • io. For, as above, AB, AF, are g "9-. alfo ABq^
7 ' BFq v CD. £D. therefore being CD is not Q.
AB, BF b fhall be uu. Wlnth^as to heWm.

in numbers, let there be AB, «?. CD, 45. CE
rr= 56. ED - 9- Make 45. 9:; z$ (ABq.)
therefoie AF — v 5. consequently BFqz=4J —
Zkzzlxq* wheretore BF rr v 20.

PROP. XXXII.

A — h To find out two medial

, , B — lines Cj D, commenfurable

Cj\ 1 ■ 11 ~ 1 only in power y comprehend-

I) . . — / j rational wluhglc

CD, /b r/^r greater
„ Cbe more in power than the leffer D by the fquareof
a right line commenfurable in length to the gi eater.
a 20. 10. "a Take A and £ p *T}- ; fo as v' Aq - Bq rx.
b 1?. 6. A. b and make A. C :: C. Bf c and A. B :: CD.
c iz. 6. I fay the thing is done.

d conftr. fox becaute A and d B are p c therefore
t zz. 10. fhall C (TV AB) be^.,f and thence alfo C~g~D.
f 17. 6. £ tnertfore D is likewile j/. Furthenriore,where-

fic. 10. 5}s A.B i C,D; and inverfely A.C B.D.v C.B j
2.4. 10. and Bq is )v. therefore (ball CO (k B^) be p?.
k 17. 6. Laftiy, becaufe Aq — Bq4 ~0- A, / fhall ^Cq
1 15. 10. - be l C. therefore, &e. But if V Aq— Bq

Aq, then fhall <J Cq - Dq be *XL C.
In numbers \ let there be A 8, B V 48 (\/ :
64 — 16) therefore C^V AB s:»y 3072.
and D = t> y' 1718. wherefore CI}r=: ? V
5308416 = V H°4«

PROP. XXXIII.

A To find out two medial

D , l\nes j), Ey commenfurable

B . — in power only, comprehend-

C ing a medial reBavgle DE9

£ -. ' fbfar that the greater D

JbaH be more in power than
the lefs £, by tlx fquare of a right line commenfu-
table to the greater in length.

a Take

EUCLIDE'x Elements: 219

a Take A and C p fo that sf Aq— Cqn. a 30. 10.
A- b take aifo B -r A and C, and make A, Db lem. 21.
c ;: D.Bd :; C. %. then D and E are the lines 10.
fought tor. c

For becaufe A and C e are p, e and B *TJ- Ad 12 6.
and C, /therefore fhali B be pe, and D y AB) geconjlr.
lhall be ^. But becaule A.D.v C.E, thcritfoicfifr-f/rA. «.
verfely A.C .; D. E. wherefore feeing A C,io.
there tore D lhall be ^) E« therefore E is u. g 22,. 10.
Furthermore, / being D. B C E. and BC is uu h 1 . 1 o.
alfo DE, equal to ic, ij a*. LaifJy, becaufe A k 14. 10.
C .v D.E. e feeing yAq Cq dl A. thereu- . I 12. ro!

V Dq — Eq t i D. therefore, &c. But if v At] m » 6 6,"
— Cq *TL A. then ^ Dq — Eq TX. Eq. n 15. 5,

In numbers, let there be A 8, C v' 48. B ^
j& then D x; V 3071. and Ev ^ 588. wherefore
t).E 2. v7 5- an^ — V iH4«

PROP. XXXIV.

lines AF, BF, incommen-
fur able in power, whofc
fqitiires added together

LU viake a rational figure,

CM-iV, £2 and the rettangle contain- * jr. 10.

ed under them mediaL b 10. 1.
a Let there be found AB, CD, f ~rj- 5 fothat c 28. 6.

V ABq - CDq ^n. AB. divide CD equally in d 12. 6.
G. c make the redanglc AEB = GCq. Upon e cor. 8. c%
AB the diameter draw the femicircle AFB, e- & V- 6.
red the perpendicular EF, and draw Af, BF. f 7- 5-
thefe are the lines required. g xo. ro.

For AE. BE d:: hA x AE. AB x BE. But BA h 10. 10.
x AE ex AFq 5 and AB x BE = FBq. /there- k ^Kf.^tf
fpre AE. EB:: AFq. FBq. therefore being AE g 47.1.
*XL EB, h AFq fhall be^ixFBq. Moreover ABq 1 conftr*
(k AFq-nFBq) I is pV. Laftiy, EFq / = AEB/— m uax*u
CGq. ?/; therefore EF~ CG. therefore CDx AB ti zz. ic*
= z EF x AB. But CD x AB n is up. 0 there- o 24, 10.
fore AB * EF, p or AF x FB is uy. JW> mf PM**£

?0 If <few. Jfc.

22Q The tenth Book of

Tbe Explication of the fame lynumlers.

Let AB be 6, CD V n* then CG = <J "f—
V j. but AE = } V <5- and EB = $ — ^ 6,
whence AF fhall be ^: i8-^ zi6. and FB 18
^- V **6\ Alfo AFq h. FBq is ?6, and AFje
FB = V 108.

But AE is found in this manner. Becaufe BA
(6.) AF AF.AE. therefore 6 AErr AFqrrAEq
-J- 3 (EFq.) therefore 6 AE — AEq = }. Put
5 -+ e = AE. then 18 6 e — 9 — 6 e — ee,
that is, 9 — ee= j. or ee = 6. wherefore e =
*J 6 and fo AE rr } ^ 6.

PROP. XXXV.

>

Gr ^

To find out two right lites AE, £2J, incommenfu*
rack in power, whole fquares added together make 'a
medial figure, and tbe reSt angle contained under
them rational.

a jz.to, a Take AB and CF ^ To that AB x CF
bepy, and V ABq — CFq -n. AB, and let the
reft be done as in the prec Prop. AE, EB are
■ ' * the line required.

For, as it is {hewn there, AEq *tl EBq. alfo
b conjtr. ABq (AEq -4. EBq) is and laftly, AB x CF
cfch. 12. h is pjr, e therefore alfo AB x DE, that is, AE x

4/A22AEB, is ff9 therefore, &c.

4 1 1

PROP.

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EUCLIDF* Elements.

PROP. 1IXVL

F

G-

E

To find out two
right lines BA, AC,
incommenfurahle in
power, whofe [quarts
added together make
C a medial figure ,

and the reSangle
alfo contained wider them medial, and incommenfu-
rahle to the figure compofed of the fquares.

a Take BC and EF a -q-, fo that BC x EF a 33.
be (jlv. and <J BCq — EFq ccl BC. and fo for-
ward, as in the prec.BA,AB, fliall be the lines "
fought for.

For (as above) BAq^TLACq. alfo BAq-+ACq
is uk. and BA x AC isuv- Laftly$BC* *TxEF,and b eonftr.
c fo BC "TLEGi likewife BC.EG<*.vBCq. BC x c 13. 10.
EG (BC x AD, or BA x AC) e therefore BCq. d i. 6.
(ABq+ACq) XL BA x AC. therefore, &e. e 14. so.

Schol

6 •

H

To find out two medial lines incommenfurahle
loth in length and tower.

a Take BC m and let BA x AC be*»f and UCL a 36. 10.
BCq (BA^ACq *make BA. H :: H. AC. thenb 33.6.
I fay BC and H are u ~g-. For BC is m. a and c 17: 6.
BA x AC (c Hq) is pv. wherefore H is alfo u. d 1 4. 10«
4 Likewife BA x AC BCq ; therefore Hq
TEL BCq. therefore,

Mere

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The tenth Book of

Htte begin the fenaries of lines irrational

by compcfition*

PROP. XXXVII.

- j_ — ■ 7/ two rational lines AB,

A B %C, commenfurable only

in power, be added toge-
ther, the whole line AC is irrational, and is called
* a binomial line, or of two names.

a hyp. For becaufe AB a ^cl. BC, thence b fhall ACq
b lem. z6* be Ti- ABq* But AB a is d, p, c therefore AC
10. is p, Wh\ch was to be dem.
cn. dcf*

10. PRO P. XXXVIII.

/ _

' - ■ - w

;A B

~- Jf taw mediallines AB,
C £ C, owty ifl power commen-
furable Jbe compounded,and
contain a rational reft angle, the wh$le line AC is ir-
rational, and called a firjl bimedial line.
a lyp. For being that AB a no. BC, b fhall ACu be
b lem. z6. un. AB x BC, pv. c therefore AC is p. Whieb
10* was to be dem.

c ii. def

io. Lemma.

A rcBangle A Cf
contained under a ra*
tionai line AB an4
an irrational line BC,
is irrational.

For if the redaa-
gle AC be affirmed,

* hP* h-> a 'hen being AB is p, b the breadth EQ
h %\M to. ihall be alfo tie Hyp.

-

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EUCLIDE'* Elements. aij

• i

PROP- XXXIX-

If two mediallincs
AB^BCyCommenf liva-
ble only in powet ^con-
taining a medial rett-
angle, be compounded^
the whole line AC fa all
be irrationaly avd is
called a fecond bime-
Q F diallme.

Upon the propounded line Dt p a make the a^.io.^
reftangle DF -ACq; £and DG ~ABq -BCq. «b 47.1.&

Becaufe ABq c xx. BCq, d therefore ABq -\ 6m
BCq, i. e. DG, TL ABq : but ABq e is^r, eC hyp.
therefore DG is pr, but the re&angle ABC isdi6.lom
taken jap} c and confcquently 2 ABC (/HF) is e 24* 10.
uv. g therefore EG and GF are p. Being alfo that f 4* 2-
DG h "n. HF s and DG- HF :: k EG. GF 5 /g *7- t
therefore EG O- GF, m therefore the whole EF iifcw. 26.
is p, n wherefore the reftangle. DF is pV, 0 there- 10.
fore J DF, i. AC, is L Which was to be dem. k * • 6.

1 10. 10*

PROP. XL, mJ7- £

n Jew. 38.

A. S C commenfurable only in

power, be added together,
waking that which is compofed of their fquares ra-
tional, and the reftangle contained under them me-
dial, the whoi* right line AQ is irrational, and is
called a Major line. a

For whereas ABq -+ CBq a isf*. and b *TL2 b fflliz*
ABC c uv 1 and fo ACq (d ABq+BCq -* 2 ABC) 10.
t *tx ABq BCq pV, / therefore fell AC be p. c hyp. ani
Which was te be 4em. z^ ia

M A ~ « e I7«

WOP.ftU/;

1

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224

tie tenth Book of
PROP. XLI.

_-n IftworigbtlinesJC,
* CB, incommensurable
in power, be added to-

gether, having that which is made of their fqudres
added together medial, and the reSangle contained
under them rational, the whole right line JB Jball
le irrational, and is called A line containing in
-power a rational and a medial reSangle.

a hyp , and j»or z re<ftangles ACB a pv, b "cu ACq + CBq

fch. ix.io.r ^ j therefore z ACBd 'rxABq. wherefore *

b fch. iz. AB is p. Which was to be dem.

I o.

c hyp. PROP* XLIl.

dl7-I0v £ D F

e ii. ief*

to.

a hyp.

tf two right lines GH, HK, incommenfurable in
power be added together, hawing both that which is
compofed of their fquares medial, and the rcttangle
contained under them medial, and incommenfurable
to that which is compofed of their fquares. the
whole right line Git is irrational, and is called a
line containing in power two medial figures.

Upon the propounded line FB/S make the reft-
b z j. I«. angles AF ~ GKq, and CFcnGHq-t HKq. Being
c 4- 2- GHq^HKq (CF) a is (jlp, the breadth CB b fliaU
be p. Alfo becaufe z reftangles GHK (c AD) a
czo. 10. js „y> therefore AC b fhall be p. Moreover be-
* 37- io. caufe thereftangle AD a tl CF, d and AD.CF
g km. j8.:: Ac. CB. e thence fliarll e AC be na. CB. /
I0- wherefore AB is g p. therefore the reitangle AF.
h 2i. def.j. e% QKq is pv > b and confequently GK is f.
Which was to be dem.

PROP.

19.

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EUCLIDE', Elements.

PROP. XLIII.

L

21$

A line oftwo names, or binomial, AB, can at one
fomt only D be divided into its names, AD, DB.

If it be potfible, Jet the binomial line AB be
divided at the point E, into other names AE,
EB. It is manifeft that the line AB is in both
cafes divided unequally, fince AD DB, and
AE U EB.

Becaufe the reftangles ADB, AEB a are ua\ a TO
a and each of ADo, DBq, AEq,EBq is act. »iSd b %A »
io ADq^DBq*andAEq^EBqarCal^i.*thcre. J 7#1
tore ADq^DBq-: AEq+EBqri.ffi AEB- z c rch < ,

ADBis/v^thereforeAEB-ADBiscVtherefore AJ/ch \z
& exceeds fxV by ft. € Winch is abfwd. J£

PROP. XLIV. •17.10.

-rf iiw^ifl/ /iwe AB, is in one point only D
divided into its names AD, DB.

Conceive AB to be divided into other names
AE, EB, whereupon every one ADq,DBq,EBq, a ?8. io.
will be a m. and the redlangles ADB, AEB, b fck Z7.
and the doubles of them fAm b therefore z AEB io

".2 A£*- f ?• * - DBq - : AEq + EBq c fck ^Z.

is pir. Which is abfwd. <f27 I0#

P PROP,

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226

The tenth Book of

PROP. XLV.

Uf C .p A fecond bimedial line
jTz A By is divided into its

names AC, CB, only at
one point C.

Suppofe there were
other names AD, DB.
Upon the propounded
75 line EF p make the rett-
* ^ ^ angles EG — ABq, and

EH — ACq •+ CBq. as alio EK- ADq-+DBq.
a 59. i©> Becaufe ACq, BCq a are *n. ; £ ACq
b 16. dw/JCBq (EH) fliall be py. c therefore the breadth
*4* to, FH is p. a moreover the redlangle ACB, Jand To
c 1$. 10, z ACB (e IG) is /ar. c therefore HG is alio p.
d 14. 10. Andfince EH is/ ^x IG, g and EH.IG :: FH.
€ 4. x. HG. A therefore FH, HG (hall be ^rx. fcthere-
f lew. z6.fore FG i*#a binomial, whofe names are FH,
10. HG. By the fame reafoa FG is binomial, and
1.6. the names of it FK, KG : contray to the 4$. of
this Book.

1

10.

k j7.

to.
10.

PROP. XL VI.

AC F 1 13 E

A Major line AB is at one point only D divided
into its names, AD, l)B.

Imagine other names AE, EB. whereupon the
a 4c, 10. rectangles A DB, AEB, are a put. a and as well
b fch. 17. ADq-DBq, as AEq + EBq are pa. b therefore
xo. ADq -h DBq — : AEq | EBq, c u e. z AEB —
c kh*$. 2. z ADB is pv. Wtticb u impoMU*
di7. to. PROP.

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EUCLlBE'i Elements.

21 J

P R O P. XLVII.

A line AS con*

J? E 3> ]B ta™*ng ™ power A

rational and a me-
dial figure is divided at one point only D into its
names AD, DB.

Conceive other names AE, ER. then both
AEq -*-EBq,and ADq h- DBq are (xa. a and the a 41- to*
re&angles AEB,ADB are pet* b theetore z AEB b feh if 4
— 2 ADB, c i.e. ADq -+ DBq AEq EBqio.
is pv. d Wl)ich is abfurd* tfth $. U

d 27. too

PROP, XL VI II.

A tine AB containing
hi power two medial rect-
angles r is at one point
o?uy C divided into its
names AC, CB.

If you will divide A &
into other names AD,
G DB. draw upon the line
propounded EF p the re-
AanglesEG— ABq,and EH= ACq +CBq,and EK
rrrADq-^DBq. then becaufe ACq- CEq, namely,
EH, a is ui/,£ the breadth FH fhall be Alio be- a 42, to4
caule z ACB,r that i$,IG,is a uv, HG b ihall be b z^. iq.
JttewiTe/. Therefore, whereas 'EH a h=l lG.and c 4. Zm
EH. IG d 11 FH. HG, thence FH e fhall be "n, d 1. 6.
HG. /therefore FGisa binomial, and the names© to. 10*
of it FH, HG. In like rmnner FK, KG fhall bef 3?. 10,
the names of it, againfi the 4?, of this Book*

Second Definitions*

A Rational line being propounded, and the
binomial divided into its names, 1 he great-
eft of whofe names is more in power than the
lefs by thefquaieof a right line commensurable
l% the greater in length; then

P » 1. if

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zzS ; The tenth m Book of

h If the greater name be commenfurable in
length to the rational line propounded, the
wjiole line is called a firft binomial line.

IL But if the lefler name be commenfurable
in length to the rational line propounded, the
whole line is called a fecond binomial.

1IL If neither of the names be commenfura-
ble in length to the rational line propounded,
it is called a third binomial.

Furthermore if the greater name be more in
power than the lefs, by the fquare of a
right line incomraenfurable to the greater
in length, then

IV- If the greater name be commenfurable to
the propounded rational line in length, it is
called a fourth binomial.

V. If the lefler name be fo, a fifth.

VI. If neither, a fixth.

PROP. XLIX.

A^C*..,.^ To find out a firft binomial

D line, EG.

zfeb. 19. E -G a Take AB, AC, fquare

10. numbers, whofe eycefs CB

b 2. km. H F is not Q. let D be propound-

10. 10. ed L b Take EF tl D, and c make AB. CB ::
c 5. km. Etq, FGq. then EG fhall be a 1. bin.
io, 10. For EF d "O- D. t therefore EF is j». /alfo
d conftr. EFq ~tl FGq. g therefore FG is alfo p. likewife
e 6Jef. \o.d be EFq. FGq :: AB.CB;: Q.not Q. /; therefore
f 6. 10. EF t~HL FG. Laftly, becaufe by converfion of pro-
g fcb. 12. portion, EFq, EFq - FGq:: AB.AC:: Q.Q. thence
10. EF/: fhall be -Q_ y' EFq — FGq. / therefore EG

h 9. re. is a firft binomial. Which was to be done*
k 9. 10. In numbers thus} let there be D8. EF 6.
1 t.defaQ. AB 9. CB 5. wherefore becaufe 9. 5 :: $6. zo9m
%o. therefore FG is ^ £0, and confequently EG is

6 ^ 10.

♦

^ _

PROP.

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t

EUCUDE'j Elements. 229

' t ' ' ' .•

PROP. L

A ....4 C.....5B Jo jM cw/ a fecond bimh

D mial line, EG.

E -G Take AB and AC fquare

numbers , the excefs of
H F which is CB not Q. LetDp^,-,^

be the line propounded p. t
take FG TX D, and make CB. AB:: FGq.EFq. r ' ■
then EG will be the line defired.

For FG U D. wherefore FG is p. Alfo EFq
hc_ FGq. therefore EF is p. Likewife becaufc
FGq. EFq :: CB, AB :: not Q. Q. thence FG is
*TjL EF. Laftly, feeing CB. AB :: FGq.EFq. and
invcrfely AB. CB :: EFq.FGq. therefore as in the
foregoing Prop. EF to. V EFq — FGq. a where- ^iMf.ifi*
by EG is a z.binomial. Which was to be done, io*

In numbers } let there be D 8, FG io> AB 9,
CB ?, then EF is yj 180. wherefore EG is ic-f
V 180.

■ * " \ PROP. LI.

A.... 4 C ..... 5 B To find out a third ii-

L ...... 6 nomial line, DF. „ N tft

G a Take AB, A C,a Jch' I*

D- ■ 1 F fquare numbers, the10-

excefs of which B C
H — -~E is not Q. and let L be

a number not Q_ next
greater than CB, viz. by a unit or two. Let G
be the line propounded p. b Make L. AB :: Go,
DEq. b and AB. CB :: DEq.EFq. then DF fhall b |. lem.
be a 5. bin. 10. 10.

For becaufe DEq c Tt Gq, d DE is p. alfo Gq.c covfir. 6.
DEq:: L. AB :: not Q. Q. e therefore G TL DE. 10.
Likewife^being that DEq e tl EFg, d alfo EF is d fch. 1 z.
p. MoreoverbecaufeDEq.EFq:: AB.CB::Q. not 10.
Q. /is DE ^T2. EF. and being that by conftr.and e 6. icw

P } of

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the tenth Bool of :

f 9. 10. of equality Gq. EFq :: L. CB :: not Q. O. ffbf g
Gfcb.iy. 89L and CB are not Like giape numbers) J there-

3

ji £t 10. fore fhall G be alfo *tl EF, Laftly, as in the
prec.Prop. v DEq — E$q u DL.i therefore

k ;.^/.48t DF is a $ bin, JW>icb was to be done.

I e. In numbers \ let there be AB>9* CB, 5« Lf &

G, 8. then lhall be DE V 96, and EF V 4f •
wherefore DF = V 4!°*- *

PROP. UI.

,M 5 C < B To owf 4 /twr*& Ji-

- i?pwid/ line, DP.

•/ti. z9, & P * JaJc? u

' # number AB, and divide

*Wf h E it into AC, CB not

fquatest Let G be the

bi Unu line propounded i. itakeDE XL. G. c and make

10: AB. rCh :: DEq. EFq. then DF fhall be a 4 bin.

c z.lenu For, as in the 49 of this Boole, DF may be

10, to, toewn tp be a bipom. ^nd alfo becaufe by conftr.

d o ro and conveifion of proportion DEq* DEq — EFq

e A/.48.:: AB. AC :: Q^npt Q. 4 fhall DE be Tx. V DE<J
K_ — EFq. e theieforeDFis #4 bin.

In numbers, let G be 8, DE> 6. t^en EF fhall
be v *4. therefore DF is 6 V

PROP. LIIL

To find out a fifth bino-
mial lme> DF.

Take any fquare num-
ber AB, whole fegments

H E AC,CB are not Q.. Let G

be the line propounded p.
take EF *n G. and make CB. AB EFq.
th$n ftajl DF be a 5 bin.

Digitized by G

EUCLIDE'i Elements, a;i

For DF fhall be a bin. as in the 50. of this
Book, and becaufe by conftru&ion, and inver-
sion DEq. £Fq :: AB. CB. and fo by converfion
of proportion, DEq. DEq - EFq :: AB. AC :: Q.
not d, a therefore fliall Dt be *xl v' DEq —a 9. 10.
EFq, b therefore DF is the s bin. Which wjs to b iJefad.
It done. 10.

In numbers, let there be G,7. EF,6. then DE
fhall be y/ $4. wherefore DF is 6 y/ 54.

PROP. LIV.

+ ••••• 5 C. ...... 7 B To find out a fixth binomial

G r — Take AC,CB, prime num-

D — - — — F bers, fo that AC CB (AB)

be not Q. take alfo any
H — — E. number fquare L. Let G be

the line propounded p. ^a j./ewi.
and make L. AB :: Gq. DEq. and AB, CB
DEq. EFq. then DF fliall be a 6 binomial.

For DF may be demonftrated bin. as in the
ji.of this Book, and alfo by reafon that DE
and EF 'tx G. laftly, likewile becaufe by con ft r. _
and converfion of proportion DEq. DEq — EFq:: '
AB. AC:: not Q. Q. (For AB is prime to AC,
b and fo unlike to it) c therefore DE "tjl <J bfch.zj&
DEq - EFq. d therefore DF is a 6 bin. Which c 9. 10.
war required. d 6Jefy9.

In numbers, let there be G 6. DE <J 48. then ia
EF (hall be y 18. wherefore DF is V 48 h- V '
2,8.

1

F 4 ^ Lemma.

#*• ■

- r

Digitized by d

2J2

if

The tenth Book of
Lemma.

. 1 . »

i.

ft 28. 6.
b $r, 1.

1

. A.

-

-~ — : — z

q Let AD U a reSangle%
" and the fides thereof AC di-
vided unequally in E ; alio
let the leffer portion EC be
euually divided in F, upon

_ _ the line AE a make the reff-

H I JC D angle AGE = awi
few Ifc points G, £,Fb
rfraip G#, £ J, FKjaraUel to
AE. c Let the fquare LM
be made equal to the reftan-
gle AH, and upon 0 MP pro-
duced the fquare MN—GI,
and let the right lines LOS>
*• -jLf * ^iroj iv r jr produced.

1 fay I. MS, MT, <rre Wangles. For by reaf0n
ih,V1Sht anSles of the fquares OMO, RMP,
•Mij-f.^ftU QAIR! be a right J jne. £ therefore RMO,
b 1$. 2. are right angles, wherefore the parallelo-

grams MS, MT are retfangles.
CZ.ax.zi. Hence it is plain that LS c z=: LT, and con-
fequently that LN is a fquare.

I. The Wangles $M MT, EK, FD are equal
$ hyp. For beeaule the rectangle AGE d - EFq. e
e 17. 6. thence fhall AE.EF :: EF.GE. /and fo AH. EK
f 1. 6\ :: EK. GI. that is by conftr. LM.EK.rEK.MN.
%[chxz£.g but LM. SM SALMN. therefore EK fcrSM
h 9. s. k — FD / = MT.
k 36. i . j. //ewce LiVw — JJ),
I 42.1. 5. Being that EC is equally divided in F, it It
tn 2. ax.i.plam that EF, FC, EC are XL .
n 16. £p. 6, If AE -c- EC, and AEnc J AEq-ECq.
o 18. Amfothen lhall AG, GE, AE, be -a. alfo, becaufe
16. ic. AG. GE :: AH. Gi. p therefore fhall AH,GI,
J? 10. 10, i.e. LM, Mty be ng.. Likewife thereupon,

7. OM

EUCLIDE'x Elements. 235

7. OM U MP. For by the Hyp. AE TL EC.

m therefore BCtlGE. q wherefore EF q 14. 10.
GE. but EF. GE :: EK. GI. r therefore EK*TL r 10. 10.
GI, that is, SM To. MN. but SM, MN :: OM.
MP. r therefore OM TX MP.

8. If AE be fuppofed *XL j AEq— ECq,it isf 19. and
apparent that AG, GE, AE, are "ttl. whence 17, ia
LM MN. tor AG.GE:; AH.GI::LM.MN.

Theft being. well confidered> wc ft) all cafily difpatch
the fix follow nig Propofitions.

PROP. LV.

If a Jbace AD he contained under a rational line
AB, and a firjl binomial line AC {AE -i- EC) the
right line OP which containeth that Jpace in power
is irrational^ and called a binomial line.

All that being fuppofed which is defcribed
and demonftrated in the next foregoing Lemma,
it is manifeft that the right line OP containeth
in power the fpace AD. a Likewile AG, GE,a hyp. and
AE are xx. therefore feeing AE is j> tl AB./<w.54-IO«
c fhall alfo AG and GE be p AB. d there- b top.
fore the reftangles AH, GI, that is the fquares c/<;0. 12.
LM. MN are j& therefore OM, MP are fe Xf-. 10.
/ and confequently OP is a binomial. Which d20.ro.
was to be dem* c S4«

In numbers, let there be AB 5. AC 4 -»- 4/ **• IO'
vrherefore the redangle AD rrio-t-^ goo = f J7- l9*
to the fquare LN. therefore OP is ^/ *S y/
j. namely a 6 binomial.

PRO P. LVI.

If a fact AD be comprehended under a rational
line Ah, and a fecond binomial AC (AE + EC) tfa
right line OP, which contains that JpaccJD in power %
u irrational, and called a firjl medial line.

Digitized by Google

z^ The tenth Book of

The forefaid Lemma of the $4 of this Book he-
ft hyp. aiding again fuppofed, then (hall OP be — V AD
a alfoAE, AO, GE are "tj- therefore fince AE h
b hyp. is j> krt AB, likewife AG, GE c ftiali bej5 Tl-
cfcb. iz. AB. therefore the reftangle AH, GI, i. e. OMq,
10. MPq. d are e Moreover OM *^-MP. Laft-

e lem. $4- ly, EF tl EC, and EC f tl AB, g wherefore
10. EK. i. e. SM, or OMP is PV. 7; Confequently OP

f hyp. iz. is a firft bme dial. Winch was to be dem.
j0m In numbers, let there be AB, 5 j and AC *J

fzo. 10. 48: -+6. then the re&angle AD — V: izoo -t- jo
j8. 10. = OPq. therefore OP is v *J 67 s v *J 75. viz.
a ftrft bimedial.

Sec SWcme $7*
PROP. LVIL

* ■

G

HI

o

a hyp. and

22. IO.

b 39. 10.

•

If a jpaceJD be contained
under a rational line AB and
a third binomial line AC
(AE+EC) the right line OP
which contains in power
the Jpace AD, is i)rational9
D and called a fecond bimedial
JN line.

As above, OPqrr AD.
palfo the reftangles AH,
GI, that is OMP, MPq
are u*. a Likewife EK
or OMP is (jut. b there-
fore OP is a fecond -bi-
medial.

X d T

In numbers \ let there be AB 5. AC J jz h
*J 34. wherefore AD is <J 800 J- <J 6oo:rOPq.
and lb OP is v ^ 4J0 -»• v ^ 50. that is a z.
timed.

PROP-

*

Digitized by Google

EUCLIDE'j Elements'.
PROP. LV1IL

B

HI

O

— If a Jpace ID be eompre-
V bended under a rationailine
F AB and a fourth binomial
AC (AE -+ EC) the right
line OP containing the jpace
AD in power, is that irra-
j£ JD tional line which is called a
Major line.
For again, OMq a *tx a Icm. 54W

is pp. c alfo EK or OMP 20. 10J
is up. d therefore OP (</ c hyp. and
AD) is a Major line. zz. 10.
Winch was to he dem. d 40. 10.

_n numbers; let there be AB $. and AC 4 -\-
^/8. then the re&angle AD is zo *\- <J zo*.
wherefore OP is ^1 zo -f- $J 200.

PROP. LIT,

1

If a Jpace JD he contained under a rational line
AB, and a fifth binomial AC, the right line OP
which contains the fpace AD in power, is that ina~
tional line, which is a line contairiing a rational and
a%medial reft angle in power.

Again OMq *T2- MPq. and the re&angle AI
pr OMq MPq is^y. a Likewife the redtangle a Asinth*
EK or OMP is fp, b therefore OP (V AD) con- pec.
tains in power pp and pp. Which was to be dem. b 41. 10.

In numbers, let there be AB 5. and AC z 4~
V 8. then the rtftangle AD ss 10 -\- *J 2wp =
OPq, wherefore OP is *j\ 10 zoo,

PROP,

*;< The tenth Book of

PROP. IX

If a fjpace AD be contained under & rational line
AB and afixth binomial AC (AE+EC) the line OP
containing the fpace AD in power is irrationaL
which contains in power two medial reS angles.

As often before, OMq -n'. MPq, and OMq

MPq is pp. and alfo the redangle (EK) OMP
a 41. re. is py. a therefore OP = V AD contains in power
z u&. Which was to be dent.

In numbers, let there be AB AC x/ iz *
V 8. therefore the rectangle AD or OPq is J
300 -+ V 200. and fo OP is ^ V 300 <J zoo.

Lemma.

□ Let a right line AB
be , unequally divided in
u S]Sc, and let AC be the
greater portion, and up-
on fome tine DE apply
the reSangles D F —

B

D1

L ABq, and DH -~ AC*.

If * and IK ^ CBq. and let

LG be divided equally in
M, arid alfo MN drawn ftraUel toGF.
a 4.2. and I fry E The re£frngle ACB is ^ LN or MF.
3. ax. 1. a For z ACB = LF.

5 V A 1 \ ?k % LG/ f<? DK C*Cq -> CBq) b <r LF
c 1. 6. (t ACB) therefore being DK, LF are of equal
d it 10. altitude, c DL ftiall be c- LG

r^* Jf AC S CB» 4 the* the ^dangle
DK be -a. ACq and CBq. b

e fc«.z6. j l/o DL LG. tor ACq -I- CBqe Xi a

? fSkVi/' PK ^ LR but DK- LF * :: DL.

f 10. I*. LQ. /therefore DL TL LG.

£ J* 6; a/5- ^te1"" DL "a. V DLq - LGq. For
lH:6* A£q\£CB * ACB' CB* th" is, DH. LN
kbyp. LJs. IK. e wherefore DL LM :: LM. IL. Ix
I 10. 10. th-jretore Dl xlL— LMq- therefore feeing ACq
» 18. 10. k xl CBq, that is, DH *ru IK. and l io DI
T£ IL. m fhall DL be TL J DLq - LGq.

Which was to be dem. *

_

f / •

EUCLIDE'i Elements, z^j

6. But ifACq he put H CB, « then JbsU DL he n 19. itw

1. <J DLq — LGq.

This Lemma is prefarasery to the following Pro-
portions.

PROP. LJCL

The fquare of a binomial line (AC-+CJB) applied
unto a rational line DE, makes the breadth DG a
firft binomial line.

Thofe things being fuppofed, which arc de-* ty?*
fcribed and demonftrated in the next preceding b 60m
Lemma ; becaufe AC, CB a are p nq., h the rett- IO-
angle DK fhall be ti ACq. c and fo DK is jv.c fch. iu

therefore DL "d- DE 0. but the redangle d n. 10.
ACB, and fo z ACB (LF) c is^y. /thereforee
the latitude LG is J n DE. £ therefore alfo *4-
DL tx LG alfo DL V DLq - LGq. from f
whence it follows that DG is a firft bingtniaLfi *?• 10-
#7;;c/; »w to be dem. h ink 60.

kiJcf.ic.

PROP. LXIL

Tie /jpjr* 0/ a jfr^ bimedial line {AC ■+ CB) be-
ing applied to a rational line D£, makes tlx breadth
DG a fecond binomial line.

The aforefaid Lemma being again fuppofed 3
the retfangle DK tl ACq. a therefore DK isa *4-*«-
f^y- b therefore the breadth DL is i DE.&2*- IO*
But becaufe the reftangle ACB, and fo LF (ic
ACB) c ispV, J (hall LG be 6 no. DE. * there-/'*
fore DL, LG are tl / alfo DL ^ DLq — <* 2-IO»
LGq. £ from whence it is clear that DG is a c **•
fecond binomial. Which was to be dem. f toa-io*

ic-

ia. *

PROP.

if

2 j 8 The tenth Book of

PROP. LXIII.

The [quart of a fecond bimcdialline (JC-+CB)
applied to a ratienalline DE makes the breadth DG
a third binomial line.
a hyp. and As in the prec. DL is p utl DE. Further-
34. is. more becaufe the reftangle ACB, and fo LF (2,
b 2^. ia ACB) a is fxu. b therefore fhall LG bej no. DE.
c lem. 6o.c Moreover DL TL LG. and alfo DL TL ^
iq. DLq — LGq. d therefore DG is a third bino-
d i*defaS.tni*l* Winch was to be dtm.
fco.

PROP- LXIV.

The fquare of a Major line (JC+CB) applied to
a rational line DE9 makes the breadth DG a fourth
binomial line.

a hyp. and Again ACq-t-CBq. i. e. DK a is fob therefore
fch. 11.1e.DL is p ~o_ DE, alio ACB, and fo LF (2 ACB)
b2i. ic. * is*tp. ^therefore LG is p "o. DE, e andcon-
c hyp. ^lequently DL "XL LG. Laftly, becaufe AC
24. 10. BC./fhall DL be xx. DLq - LGq.£ whence DG
d 2$. 10. is a fourth binomial. Winch was to be dem.
e 13. io.

f lent. 60. % PROP. LXV,

to. The fquare of a line containing in power a ra+

tf ^fitf rtwMi a re& angle (AC-\ CB) applied

10. j rational line DE makes the breadth DO a

a 22. iq. JS*' tinomiaL

b it, 10 ASain> DK is ^v. a therefore DL is p -c DE*
c , , ' . * alfo LF is fo. b therefore LG is I ~gl DE. *
£ X. therefore DL U LG. J likewife DL

d/ra. i0# therefore DL U LG. J likewife DL *U- V
t0# #DLq— LGq. e and fo by confequence DG is n
« S.def.^8 binomial. JPiif & fo faf*

^ PROP. LXVI.

The fquare of a ime containing inpower two medial
teft angles (AC-* CB) applied to a rational line DE9
mikes she breadth DG a Jixth binomial line.

At

Digitized by Google

EUCLIDE'i "EUmtnts. 239

As before, OL and LQ aie i *TX DE. Buta fop
for that ACq -»• CBq (DK) a 'n.'ACB, i and b il 10.
fo DK -n LF 0 ACB) and alfo DK. LF e ::c\ 6.
DL.LG. d theretore fliall DL be TL LG. ed 10.ro
Laftly, DL "tl V DLq - LGq. /by which it c /«. 60.
appears that DG is a fixdh binomiaj. io.

Lemma. f

C

i B

F E
D 1

Let JB, DE he "D-. and make JB. DE ::
AC. DF.

I fay 1 . AC XL DF. as appears by to. 10.
alfo CB -n. FE. a becaufe AB.DE CB.FE. - ,„ -

z. AC. CB .v DE.FE. ForACDF :: AB. DE .•: 9' S*
CB. FE. therefore inverfely AC. CB .•: DF. FE.

j. JhereSangle ACB tl DFE. For ACq.ACB
h :: AC. CB c :: DF.EF.v DFq. DFE. wherefore 5 . g
by inverfion ACq.DFq.v ACB. DFE. therefore - ilrl.
being ACq -o. DFq. d fhall ACB be n. DFE. a jTv*

4. ACq -h CBq -a. DFq ■+ EFq. For becaufe
ACq. CBq e DFq. FEq. therefore by addition* x
ACq - CBq. CBq DFq -|- FEq. FEq. there- * '°*

Digitized

240 72* tenth Boik of

PROP. LXVII.

* «

ne D£, commen-
ce in length to a ii-
nommlline (AC->CB) is
it jelfa binomial line yand
of the fame order.
Make AB. DE :: AC. DF. a then are AC,DF
a km. 66. TL a and CB, FE tl. whence being that AC
10. and CB £ are p ng. , c thence DF, FE p there-
b £y/>. fore DE is a binomial. But for that AC?CB a ::
c lem. 66. DF. FE. if AC n or U V ACq - BCq. d
io.andfch.then in like manner DF "u or , V DFq —
12. 10. FEq. alfo if AC XL or 'n. j> propounded, e
d tj.io. then fhall DF be tx or ltl p propounded. But
c xz. 1 a. if CB or 'tl p. likewife FE "a. or * ex p.

14.10. If both AC, CB, "a. p. rthat is, whatfocver
fbydeftf. binomial AB is, DE (hall be of the fame order,
le. Which was to be dem.

PROP. LXVIII.

a it. 6.
b
10

jf lineDE commenfurabte in lengih to a bhnedial
line (JC-+CB) is alfo a Hmedial lme$ and of the
fame order.

Make AB. DE AC. DF. * therefore AC

/ * DF and CB tl FE. therefore feeing AC and
jew. 00. CB e d alfo DF< aijd F£ ftaU be ^

* W . for that AC e "3- CB. e therefore FD tj. FE.
a ?2tr /therefore DE is z u. Wherefore if the redan-
tin 10 gk ACBbe^r. becaufeDFEiTDLACB^Jikc

f !& Ic! wifc DFE ** h if that bc * thi* fta11
K h. iL bc ^r too# * ^^at ^ w^cthcr AB be 1 bimed.
i£ * or 1 bimed. DF fhall be of the fame order

10*

PROP.

Digitized by Google

EUCLIDE'* Elements'. 241
PROP. LXIX.

;

A 1 B Aline DE commensurable

1 C to a Major line (AC -¥ CB)

D ' ■ 1 1 E is it felf a Major line.

F Make A£. DE .v AC.

DF. Becaufe AC a *TJ- CB, b thence DF no. a hyp.
FE. Alio ACq-^CBq a is ^V. and fo being DFq b /ew. (5i.
H-FEq b ra. ACq+CBq, c alfo DFq+FEq is/,, to.
laftly, the reftangle ACB a is ^. d therefore the c M iz.

*e™gIe PFE is (becaufe Df E is * tl to.
ACB; e wherefore DE is a Major line. d z^f 10.

„ _ ^ e 40. 10.

PROP. LXX,

. A line DE commenfuralle to a line containing
m power a rational and a medial reSangle (AC-l
CB) is a line containing in power a rational and a
medial reSangle.

Again make AB. DE :; AC. DF. Becaufe AC
i> £B,.*alfoDF V FE. Jikewiie becaufe a ht
ACq-| CBq a is ^, c theretore DFq+FEq ftall b t£ ~
be f». laftly, becaule the reftangle ACB a is .V., ' ' 66'
ialloDFEis ft. Therefore DE contains in °" Tn
power 'ft and ^ Which was to be dent. J r% '

10.

e 41. 10.

A line DE commenfu-
rable to a line contain-
ing two medial redangles
in power {AC -J- CB) is

■• ' • ; alfo A line containing in

power two medial reSangles.

■ Divide DE, as in the prec. Becaufe ACq a a /;y*.
CBq, b thence fhall DFq be *T2- FEq. alfoforb&w \tf

t?t r 111 1|kc«lann« becaufe ACB* is^alfo c 24. 10
VHis Laftly,becaufAACq^CBq ^ ACB.d it \o.

Q f fhaU

Digitized b

24* 7&e tenth Book of

e 14.10. e Aall DFq^FEqbeTLDFE- /from whence
f 41.10, it follows that DE contains in power 1

Which was to be denu

• v * ••• • .. a

PROP. LXXIL

* * *•

H

Tf a rational reS-
angle A, and a me-
dial B, be contpofed
together, thefe four
irrational lines will
be made \ cither a
binomial, or a firfi
bimedial, or a ma-

jor, or a line containing in power a rational and a

medial re& angle.

Namely, if Hq~A I B. thenH fliall be one of

the four lines which the Theorem mentions. For
ara>.i<5.6.uPon CD the propounded p, a make the re&an-
b z. ax.i. gte CE A, and FI = B. b and fo CI Hq.
c 11. 10. Whereas then is A pv, likewifeCE is py. c there-
d 13. 10. f°re trie latitude CF is p tl CD. and becaufeB
e 1 3. ic. is v** FI fhall be^r. d therefore FK is J tu~
f 37. 10. CD. e therefore CF, FK are p XL. and (o the

0 1 5 whole CK/is binom. wherefore if A cr B, i. em
h r W.48.CE cr FI, g then CF cr FK. therefore if CFtx.
io. V CFq - FKq, * likewife CK fliall be a 1 bin.
k 55. 10. and confequently H -= ^ CI k is a bin. If CF

1 ±def.fi. be luppofed 'n. V CFq - FKq, / then fliall CK
10. be a 4 bin. wherefore H <y CI) m is a major
m 58. 10. line. But if A -71 B, g then fliall CF be -3 FK,
niMf.48. confequently if FK tl x/FKq - CFq, w then
10. lhall CK be a z bin. 0 wherefore H is a firft zu.
o 56. ic. laftly, if FK ^n. V FKq - CFq ; / then CK
P tJef-& lhall be a fifth binom. ^ whence H fliall contain
10. in power fv and pr* Which was to be dem.

q 59. 10.

PROP*

-

Digitized by Google

EUCLIDE'j Elements:
PROt. L^XIH.

Ml

IftwomediaheB-
angles J, £, incovi*
menfurable to on*
another be compofed
together, the two re-
maining irrational
lines a) e made,either
I a fecond bimedial^ot
a line containing in power two medial ie3 angles.

As H containing in power A Bis one or the
faid irrational lines. For upon CD propounded
f draw thereclangle CE=A,and tlrrB. whence
iJqrrCI Therefore becaufe CE and FI a are 44. a tyh
I the latitudes CF, FK, ftiall be p ^ CD* alfo b 25. 10.
becaufe CE a 'tl FI, and CE. FI c n CF. FK, c 1. 6.
d therefore CF U FK. e therefore CK is a 5. bin. d 10. tOJ
namely, if CF -dl V CFq - FKq. whence H e iMffi.
— A/Clf (hall be a fecond z ^. But if CF H f *7- j o.
V CFq — FKq, ^ then CK fhall be a * binom. g6.<fc/.4&
/; and consequently H contains in power z (ut.io.
Which was to be dem. h 60. 10.

Rtoe begin the Senanes of tines irrational by

SubtraSion.

PROP. LXXlv.

v jT ' " If from a rational line DF a

D £ F rational line DE, commenfura^

ble in power only to the whole
DF, be taken away, the refiiue EF is irrational*
and is called an Apotoyie or refidual line.
, For JEFq a ^ DEq; b but DEq is fp j c there- a lem. 2&
fore EF is p. tylmh was to be dem, % .10.

In numbers, let there be DF, I. DE^/j. thenb hyp*
EF fhall be z p* y 3, cio.&iu

, . dtf l©«

Q,* PROP,

Digitized by

244 the ienA Book

■

prop, ixxv.

t> E ¥ If from a medial line DFy a

> medial line DE commenfurdble

ontyin power to the whole DFr
and comprehending with the whole DF a rational
reSangle be taken away, the remainder EF is irratio-
nal, and is catted afirft refidual line of a medial.
a fch. 16. For EFq a ""CL to the reftangle FDE. thereT
to. fore feeing FDE b is fa c BF fhaii be Winch
b hyp. & he dem,

c 20. and In numbers, let DF be v ^/ 54, and DE v tft
1 1 . def.io. therefore EF is v <J $4 — v j 24.
J PROP. LXXVI.

D E F If from a medial line DF, A
~ ' ■ ■ , medial line DE be taken away

- being incommensurable only iri
power to the whole DF, and coinprehending together
with the whole line DF a medial retlangle, the rfc-
mainder EF is irrational, and is catted afecond re-
fidual of a medial line.
a jqfm Becaufe DFq and DEq k are ua Tl,9 h there-
b 16. ie. fore fhallDFq DEq be n. DEq. c wherefore
c 24. 10. DFq DEq is alfo the reftangle FDE, c
d cor. y.r.and fo 2 FDE, a isfxv. therefore EFq (d DFq-n
e 17. 10. DEq — 2 FDE) e is fa wherefore EF is
Which was to be dem.

In numbers, let DF be v J 18. and DE v ^
8. then EF v <J 18 — v V 8.

PROP. LXXVIL

If from a right line AC

A B C taken away a right line AB hi-
ring incommenfurable in power to
the whole BC, and making with the whole AC that
which is compofed of their fquares rational, and the
rectangle contained under them medial, the remainder
a hyp. BC is irrational, and is catted a Minor line.
b fch. I2> For ACq-^ABq a is fa but the reftangle ACB
10. a is uv. b therefore 2 GAB ^ ACq ABq (2 c
c 7; 1. r • CAB

Digitized by Google

EUCLIDEV JZhmenis. 54f

CAB-»-BCq.) d therefore ACq+ABq TL BCq. *d 17. 10.
therefore BC is }. Which was to he dem. e tl\ jjf

In numbers, let AC he y/: 18 -fa V i«8 ; AB 10. '
r8 — V 108. then BC is ^8 -4- V: 108 ~
v: 18 r- V 108.

PROP. LXXVIH.

// from a right line DP
he taken away a right hne
DE, heing incommenfuralle in power to the whole
line DF, and with the whole DF making that which
is compofed of their fquares medial, and the reft angle
contained undo- the fame lines rational, the line re-
maining EF is irrational, and is called a line making
a whole fpace medial with a rational fpace.

For 2 Ft)E a is pp. I aad DFq+DEq is fxp. c a Jjyp. and
therefore 2 FDE tl DFq+DEq d (1 FOE +EFq)/dh 12.10.
e therefore EF is p. Which was to be dem. b Iryp.

In numbers, let DFbe^/: 216 -+ <J 72; DE c fch. 12.
V': V i*6 — V 7** therefore EF is V 1 1<* IO-
V 7* — V- — V 7** ^7. 2.

e fch. it.

PROP. LXXIX. 10.&11.

defio.

|i n i l away a rigfo DE ixcommenfu-

rable in power to the whole DE, and
which together with the whole makes that which is
comfofed of their fqutres medial, *nd the teSangle
contained under them, cdfo medial and uicommenfu-
tablc to that which is compofed of their fquares, the •
remainder is irrational, and is called a line making
s whole fpace medial with a medial fpace.

For 2 FDE, and FDq-+ PEq a are fut \ h there- a hyp. and
fore EFq (c DFq+DEq — 2 FDE) is j>v. d and 24. 19.
So consequently EF is p. Which was to he deny b 27. 10.

In numbers, let DF be V 180*1-^60. DEcw.7.2.
V: V *8o - V 60. then EF fhaU be v: V l8od- lr- <^f*
TT V 60 — V 180 — V *Q* *«•

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Tie tinth Book of
Lemma*

• • — -

— r

If there he the lame exeefs between the fit ft magni-
tude BGr and the fecond C (MG) as is between the
third magnitude DFand the fourth H\EF\) then
'alternately, the fame exeefs Jball he between the firfi
'rndgnitxide BG and the third ! DF, as is between the
fecond C and the fourth H.

ft lyp* For becaufe that a to the equals BM, DE,
are added the unequals MG,EF, that is, C,H j

pi<.hx.i, the exeefs of the wholes BG, DF, b fliall be
equal to the exeefs of the parts added C, H.
JPhich was to U dem. '

. • Coroll.

Hence, Four magnitudes Arithmetically pro-
. portional, are alternately alfo Arithmetically
proportional. /

P$0P. LXXX.

B D G To an Jpotome or reft-
A -1—1 dual line AB only one ra-

tional right Hne BCj be-

JW. iof ing commenfurable in power only to the whole JB,
iz- 10. U congruent, or can be joined,
f cor. 7.2," If it be poffible, let fome other lineBD be ad-
d lev. 79- ded tb it } Jx then the reftangles ACB, ADB, h
to. and fo cdnfequehtly the double of them are ua.
e byy, <wd wherefore feeing ACq+BCq— i ACB<r=ABqc
X7< io. b'ADq'-f DBq X ADB. therefore alternate!*
ffch. iz. AGqVBCq-iADq^DBq^xACB-:iADa
%o. 4 But ACq^BCq;-:ADq^DBq« is ^./therefore

g *7- IP- i ACB i ADB is p. Which is abfuii.

T - * f • ' ' PROP.

» •

v

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EUCLIDE'i Elements. 247
fROP. LXXXI.

To a firft medial refidual line

A B D G JB only one medial right line

BC, being commenfurable only,
in power to the whole, and comprehending with the
whole line a rational reUangle, can be joined.

Conceive BD to be luch a line as may be
joined to it ; then becaufe ACq and BCq,a.swell
asADq and BDq a ate M-CL.b alfo ACq -■- BCq,*
and ADq+BDq fhall be uu. c but the tedangjes 0 lo-ana
ACB.ADB, <i and fo z ACE and i ADB ate P*.M- *°-
e therefore t ACB - • * ADB, / that is, ACq -\-c. W;
BCq-:ADq-t-BDq is P'v. g W** " «V»A

PROP. LXXXU. « M &
Aft c t* f»'o « media I' o> .
^-j-D y-yefidual Une AB onlyf 7- *• an«

BC,commenfurable onlyZ »7« I0«
i» power to the whole,
and with it containing
a medial reSangle, can
be joined.
fi OB If it be poffible, let
fome other line BD be
added to itj and upon EF P% make the jangle - z

-O, EF. e Further, the reftangle ACB / and to ». f«-
i ACB (KG) is p,. d therefore KH is alfo B, i g'««- r* ■
EF. laftly,becaufe ACq +BCq (EG) g *tl * ACB >°;
(KG) and EG. KG 6 EH. KH. k therefore J ,- «-
BH-ELHK. / therefore EK is a refidual line * 10* *
whereto HK is congruent, by the fame reafon 1 7*»'*

Q.4 ^

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The tenth Book of

alfo lhall KM be congruent to the fatf EK.»bic6
is repugnant to tie 80. Prop, of this Booh

PROP. LXXXIIL

■ To a Minor line JB only

A B DC one right line BC can te join-
ed beittg irtcommenfurahle i*r
power to the whole ; and making together with the
whole line that which is compofed of their /wares
rational, and the reSangle which is contained under
them medial*

Conceive any other BD to be congruent to
it : therefore whereas ACq -\- BCq, and ADq-*-

a pp. DBq a are fa their excefs (2 h ACB -s 2 ADB)

b lent. 97-cis }y, Winch is ahfurd-9 becaufe ACB and ADB

10. are ^ by the Hyp/

cfch. 27.

«* PROP. LXXXIV.

d 27* 10.

* Zfa/o a line (JB) making

A B D C wifA rational Jpace a whole

fface medial only one tight
line BCcan be joined, leinginconimcnfurable in power
to the whole, and making together with the whole
that which is compofedsof their fquares medial,
and the reSangle which is contained under them,
rational.

2 hit Suppofe Tome other BD to be congruent alfo
b tch 12 r° lti££*n the reftangles ACB, ADB, b and
U fo * ACB ; and 2 ADB are ^. therefore z ACB

lent to 7l.Z ADB> rthat is» ACq^BCq -:ADq^BDq

d /ci. 27, AD(* + BD9 *|e ^ fay Hyp,

c
1©.

Digitized by

EUCLIDEi EUmntZ

PROP. LXXXV.
"B C. D To a lint M, which

with a medial /pace
makes a whole /pace
medial, can le joinci
only one right line BC9
incommensurable in
power to the whole, and
making with the whole
both that which is com-
F Jofed of their ftntares
medialyand the red angle
which is contained under them medial and incommcn-
furable to that which is compofed of their fquates.

Thofe things being fuppofed which are done
and fhewn in the 82. Prop, of this Book ; it is
dear that EH and KH are p Tl EF. Befides,
being that ACq+CBq, that is, the rettangle EG,
/1 is ,xl ACB. % and lo EG *xl 2 ACB (KG;)a typ;
and EG. KG* .vEH.KH; lhallEH be *tl KH.b 14.10:
therefore EK is a refidual line, and the line con- c l« 6. *
gruent to it is KH. In like manner may KM
be fliewn to be congruent to the faid refidual
EK, againft the 80. Prop, of this Book.

Third Definitions,

A Rational line and a refidual being pro-
pounded, if the whole be more in power
than the line joined to the refidual, by the
fquare of a right line commenfurable unto it
in length \ then

I. If the whole be Commenfurable in length
to the rational line propounded, it is called a
firft refidual line.

II. But if the line adjoined be commenfurable
in length to the rational line propounded, it is
called a fecond refidual line.

III. If neither the whole nor the line adjoined
be commenfurable in length to the rational line
propounded, it is called a third refidual line.

m More*

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7U tenth Book of

Moreover, if the whole be in more power tha*
the line adjoined by the fcjuare of a right
line incommenfurable to it in length, then

IV. If the whole be commenfurable in 1 ength
to the rational line propounded, it is called a
fourth refidual line.

V. But if the line adjoined be pommenfurable
in length to the rational Una propounded, it is
a fifth refidual,

VI. If neither the whole nor the line adjoined
be commenfurable in length to the rational line
propounded, it is termed a fixth refidual line***

PROP, LXXXVI, 87, 88, 89, 90, 9r.

To find out a firft\fccondjhiri9
fourth, fifths and fixth refidual
line.

Refidual lines are found out
by fubdufting the lefs names
or parts of binomials from
the greater, ex. gr. Let 4 -h ^ ao be a firft bi-
nom. then fhall 6 — V io be a firft refidual. Sp
that it is not neceffary to repeat jnorc concern-
ing the finding of them out.

Lemma.

T p y £jj Let AC he a reSangle con-
tained under the right lines
AB9 AD. Let AD be drawn
forth to £, and DE equally
divided in F. and let the relf*
angle AGE be ss F£a. andm
5ttH tht H&anglw AlfiKfH, befi-
tflifbed. Wen let thefquareLM
r— AH be made, and the fquart
Qm-Gl ; and the lines NSR%
OST. produced.

I lay 1. the reftangle
= LM -r NO = TOq -h
SOq. which appears by th$

urn

uigitizf

ed by Go(

EUCLIDE'/ Elements. mfA

t. The reSangle DK — LG. For becaufe the
teftangle AGE a FEq. £ thence are AG, FE,a conftr.
GE, andfo AH,FI,Gl ff, that is, LM,Fl,b 17.6.
jsjq Ji* bUt LM, LO, NO ^ are -H; jhereforec r. 6.
FI = VlO / DK # NM. • d fcb.v.6.

/fcw*, k DK FI-LM NO- LOe 9. 5-

£ J/ AE -q- D£, ^£ tx V AEq-DEj,h \6. 10.
i *fen JbaU AG, GE, JE be~c. k 18. and

6. Alfo, lecaufe AE I Tl Di, w thence JbaU 10. 10.
jfi, PEbe *XL. » /iwi /a AIy Fly that jj, Litf j 1 hypm

AO tfwi LO are L^CL' m 1

7. iertftf/e jfG * TX G£, -i//, GI, that n 1.6. aai

8. in* lecaufe AE I 'tl Di, 0 therefore JbaU* hefore
FE> GE be^CL, n and Jo the reH angle FI ^ GI,0 14. 10.
/fort ix, JLO 'XL M). wherefore Jeeivg LO. NO p :: p 2. 6.
75. SO. a therefore JbaU IS, SO fa tx. q i*, io.

' o. J/^£ be put -T^ j AEj- DEq, r them 19.10.gf
fiailJGjGE, AE be "a.. 17. 10.

10. / Wxrefore thereBangks AH> GJ, ffo/ is>{ i.6.<wd
TO?, 60jf ^tf/f TU 10. 10.

fROf,

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7li tenth Booh of

PROP, xcn.

!P !F <SE * /j^e ie contained
under a rational line AB, and
a firft refidual line AD (AB
— DE) the right line TS,
which contains the fpace AC in
power, is a refidual line.

b ii. i©.
c 20. io* <£
d lent. ox
10.

C KHI- Ufethe foregoing Lemma
p for a preparatory to the de-
monftration of this Prop.,
0 Therefore TS — J AC
Alfo AG, GE, AE, are -m.;
therefore fince AE tr a AB
f, J alfo AG and GE fhall
be tx AB. c therefore the

~p -fcf uc c inererore tne

* « reftangles AH and GI, that
is, TOq and SOq are fa. d Likewile TO, SO,
C74.ro. aJe p "9-. e and confequently TS is a refidual
lke# /f7;iri w<zr to be dem.

• • • •

prop, xciir.

*

See the (receding Scheme.

. V *fpace AC he contained under a rational line
AB and a fecond refidual AD (AE - DE ) the
right line TS, containing the ftadc AC in tower ,
U a irfi medial refidual line. • »
^ a *2ain> by the foregoing Limp*, AG, GE,
a typ. AE are tl. therefore a fince AE is p ^tL AB,
b 15.10. ialfo AG, GE, fhall he £ AB. c therefore
c 22. 10. the rectangles AH,GI, that is, TOq, SOq are M
d lent, 74. <* like wile TO TJ- SO. Laftly, becaufe DE e

1QL *P- ABi* ^the ri8ht anSle Vi> and1 the half
« bjt. thereof DK or LO, that is, TOS fhall be fa. g
1 zo. 10. from whence it follows that TS U AC) is a
* 7* io. firft medial refidual. Which was to be dem.

PROP.

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PROP. XCIV,
See Sctetne $i.

' , If a fpace AC he contained under actional line
AB and a third refidttal JD (AE - DE) the rial*
line TS containing in power the fpace AC is a fo»
cond medial refidual line'. J

As in the forrnfer, TO and SO are p. There-
fore becaufe DE a is f 'r l AB, & the reftangle a %.
DI, c and fo DK, or TOS, Ihall be pv. d there- b zu ioC
TS = V AC is a fecbnd medial refidual. c 24, 10.
Which Was to he dent. d 76. 1*

PROP. XCV.

1

See Scheme p£
If a fpace AC he contained under a rational Une
AB and a fourth refidual AD (AE—DE) the right
line TS containing the /face AC in power, ' is A
Minor line.

As before, TO a T}. SO. Therefore becaufe a tern. $u
AE b is f tl AB, c {Hall AI (TOq -+ SOq) be 10.
fy. but, as before, the reftangle TOS is uv. d b hyp.
therefore TS = V AC is a fliinor line. Which c 20. 1*
vastobedem. 'd 77.10.

PROP. XCVL
Scheme 92.

if /^tfrt JCfre contained under a rational line
AB and a fifth refiduMl AD (AE - DE) the right
Une TS containing in power the fpace AC, is a
line which makes with a rational fpace the whole
fpace medial.

For again TO ^ SO. therefore fince AE a is a hyk
I -xl AB. h alio AI, that is, TO4+SOU, ffaall b 22. to.
be ^. But, as in the 93. the reftangle TOS isfo. c 78. to.
c whence TS — <j AC is a line which with fo
makes a whole pr< Which was to he dem. ■

PROP,

Digitized by

, ~ «•

**4

7& BwA of :

PROP. XCVIL

under a rational line JB9 and
a fixth refidual At) (ifj? —
DE) the right line TS con-
taining in power the fpace JC,
is a line making with a

q jl dial reSangle, a whole fpace
x medial*

As often above, TO *TJ.
>SO. alfo, as in 96. TOq ~^
SOq is pv. but the reft angle
TOS is pr, as in 94, a Laftly,
tOq ■+ SOq TL TOS, *
. , therefore TS sfc V AC is
R Ha line which with makes
a mrhole ^ F&icA v*i to he ion.] 1

*cor.x6.6.

Si

^ g l7jpo» right line DE *

-vi *fo reS angles DF~

SJL-i^^, ^ bH-JCqy and

IXzzBCq. and let QL.hc
hifeSed in M, ana the line
MNdrawnparaUelto GF.
Then 1. The reSangle

m -p ; u irj-DKis—JCj+BCq. as the

W 1 H OX.conpuaion manifefts. , >

a cowffr. 2. reSangle ACB = GN ©r MK. ForDK
b 7. 2. « = ACq -♦• BCq b = z ACB -h ABq. but ABq
c j. ax. 1. d DF. therefore GK c — z ACB. and d con-
a 7. dw< 1. fequently GN or MK = ACB.
e 1. 6. ?• fc& r«,?We DIL = MLq. Fpr becaufe
t n. 6. ACq. ACB e :: ACB. BCq, that is, DH, MK «
MK. IK. e thence is DI.ML .•; ML.IL . /there-
fore DIl^MLq.

EUCLIDE'i Ekmtnts.

g itf . IQ.

4. Jf^fC fottie* ^ BC, then DKJbai be
JCq. For ACq -+ BCq (DK) £ TL ACq.

5. Likewife DLn. V DLq-GLq. Forbecaufe
DH (ACq) -cl IK (BCg) h thence ihall DI be h ia ia

X lL. k therefore J DLq-GLq -a. DL. k ia id*
tf. JZ/o DL TL GL. For ACq + BCq ^tl /l few. *&
* ACB. that is, DK TX OK. w therefore DL ia
4tl. GL. m io. io*

7. £k* i/ AC be taken ^ BCf n then DL Jballn 19.10*

PROP. XCVIIL

Tie fauare of a refi*
iualIineAB(JC-BQ
applied to a rational line
Z)£, makes the breadth
DG afirft tefidualline. ^

Do as is injoined in h (JnL 07
the Lemma next pre- Ia w*

alfo DK (ACq+BCq)fhall be tx ACq. £ there- j ' vn
fore DK is JK. <* wherefore DL is 8 tl DE. e°"' IS
Likewife the reftangle GK (t ACB) is pr. /! 7 ,0
therefore GL is 5 Tl DE. £ and confequently e V,
DL TL GL. A But DLq "o. GLq. k therefore * J*' , °"
DG is a refidual, / and that of the firft order r /-A
(becaufe « AC T3- BC, and therefore DL "D- ,/
V DLq- GLq.) Whisbwattohedm. v i 74.10.

10.

m /ejn. 97.
10,

*RQP,

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7U tenti Book if
■ •
PROP. XCIX.

$ee the following Scheme.

The fqutre of a firft medial refidual line AB {AC
* BC) applied to a rational line DE, makes the
breadth t)G a fecond refidual line.

Suppofingthc foregoing Lemma i becaufe AC
and BC a are ^ "q., h thence (hall J

and BC a aFe*^,* thence fliall DK (ACq..
t> lem. 97-BCq) be TX ACq. c wherefore DK is pv. d
io. therefore DL is p 12- DE. e alfo GK (2 ACB)
c 24. 10. is therefore GL is p ncu DE, £ wherefore
d 23. 10. DL uix GL.. h But DLq GLq. k therefore
e hyp. andQQ jsa refidual line : and becaufe DL is TL^
fch. 12.10. £)Lq _ GLq* * therefore fliall DQ be a fecond
£21.19. refidual. ifHrnh wasto be dem.

f 1^.10.
fch. 12. PROP. C.

xo.

£74-10- _ The fquare of a fecond

1 fern. 97. A ^ C weiw/ r^Mnfli /in* A3

m 2. A/, -^f J \ational Ime makes

8s- 10. J I 1 , the breadth DG a third

refidual line.
Again DK is /uy. a

a 2}. 10

b lem. 16. « — -kt uir wherefore DL is p "a-

iq. 16 F N ^^DE. alfo DGK is ur. *

c 1.6. aiid whence GL is p H DE. > likewife DK H.
10. 10. GK. c wherefore DL, ixl GL. d but DLq
d fch. iz. GLq. e therefore DG is a refidual line, and that
10. of / the third order, g becaufe DL "O. V DLq

e 74. to. — GLq. Which was to be dem.

fl.def.Si. ■ ;

10. PROP. CI.

lem. 97.

iq. See the foregoing Scheme.

t

Tbef quart of a Minor fine AB (AC—BC) afflbik

to

ngmzeo

y Goo

EUCLIDE'/ Elements'. 257

to a rational line DE, makes the breadth DG &
fourth refidttal.

As before, ACq BCq, that is, DK is iv. a% zx. io.
therefore DL asp TJ_ DE. but the rectangle ACB** hyp
and fo GK (z ACB) * is py. b wherefore GL isb t\ toi
i U DE, c therefore DL U GL. 6bufci?'io.
DLq tx GLq. and becaufe * ACq Tx BCq, ed rc'h iz

thenceftall DL be '-n-VDLq-GLq./there-ro. ' *
fore DG has the conditions required to a fourthe Um. 07.
refidual. Winch wot to be dim. I0> V1

PROP. CIL [t[^

See Scheme 100.

Thefquare of a line AB (AC- BC) which makes
with a national jjtace the whole Jfrace medial, applied
to n rational line DE, makes the breadth DG a fifth
refidual line. JJ

For, as above, DK is a wherefore DL is pa io*
'til DE. alfo GK is fr. b whence GL isp io.
DE- c therefore DL ltx GL. d but DLqncLGLq c lh IO-
Moreover DL* %/ DLq — GLq. wWeforV fih. iz.
DG / is a fifth refidual; Which was to be dem. 1 °*

e tem. $j.
PROP. CIIL io.

f $-def.8s.
See the lajl Scheme. 101

s

The fquare of a line AB (AC—BC) makingwhh
a medial foac* the whole f pace medial, applied to a
rational line DM, makes the breadth DG d fixth )e.
fidual line.

^ As above DK and GK are ^ ; a wherefore a ros .
DL and GL are p tl DE. alio DKb XL GK; b bp. and
c whence DL TL GL. d therefore DG is a rt-^w.oj.xo*
fidual. b And whereas ACq *xl BCq. and fo DL c iQ. 10.

4 DLq — GLq, e therefore DG fliall be a d 74* 10.
fixth refidual* Which wasty be dm. e 6Jef.8$;

10*

* 5ROP4

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% j 3 Tie tenth Book of

■ •

PROP. CIV.

1 . G A fight line DE com-
•B wenfurable in length to a

J) 1 F refidual JB {AC — BC) is

E jt felf alfo a refidual) and

of the fame order.

• Lemma.

Let AB. DE :: AC. DF. and AB TL DE.
I fay AC -1 - BC no. DF EF. For AC. BC
a :: DF. EF, therefore by addition AC -+ BC.
. BC :: DF EF. FE, therefore by inverfion AC
a leni. 66. ^ BC. DF -\- EF :: BC. EF. a but BC tl EF.
i o. b therefore AC -\- BC TL DF -J- EF. Which
b io. io. waf to be devt.

a 12. 6. a Make AB. DE :: AC. DF. b therefore AC+
b /w/.iog.Bc "C- DF -+■ EF. therefore feeing AC-\ BC c
io. is a binomial, d DF EF ihall be a binomial
c too, ancf of the fame order, e wherefore DF —

d 67. 10. £F is a refidual of the fame order with AC
e iy ^/-BC. Which was to be dem.
85. ioi

PROP. CV,

•

A B C A right line t)E covwienfu-
1^— _ table io a mtdial refidual line

T-*— JB (JC-BC) is it felf a

D E F medial refidual, and of the fame

order.

* iz; & Again a make AB. DE :: AC. DF. b whence
blem* 103; AC BC xl DF - EF. r therefore DF EF
10. is a bimedial of the fame oTder with AC-»-BC,
c 6& 10. d and confequently DF - EF lhall be a medial
d7J. and refidual of the fame order with AC-~.BC. Winch
?6. icn jva* *o demonjlratedi

PROP*

r

-

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EUCLIDE'i Elm'tntA if*

P R O P. CVI.

* ■

A ii C A right line BE contmen*

_ 1 furahle to a Minor lint AB

1 (AC- BC) is ~itfelf alfop

D E ¥ Minor lint* .

Make AB. DE :: AC. DF. a then is AC+BC* «*.I0J.
Tl DFh-EF. But AC -h BC * isa Major line -
c therefore DF-+ EF is alfo aJitajor line \ d and &
confequently DF ■— EF is a Minor line. Jffc**S for
was to h 'dm. d 77. to.

pro?, cvu./

A C y(g*W Ziwe Di? commenfu-

m y~ — ralle to aline AB (AC — EC)

"jT"-" 4; ^ which makes with d rational

[face rfje vbole J face medial,

is it. [elf alfo a line making with a rational [face

the whole [pact medial.

. For, accordingly as in the former, we may

fliew DF -hEF to contain in power h and uy. tf- TA«

whence DF ~ EF is a line making, &c. * '* f*

PROP. CVIII.

A B C j# ri^g-fa /iwe DJ? commenju-

; — -1 ™&/<? /o * toft AB (AC ~BC)

1~ irZwi jri/Zr # medial [face

D E F wtf fee* Me, w/;ole /p* r£ medial%

is it [elf a line making with
a medial [face the whole [face ihediaL

For according to the preceding DF EF
lhall contain in power 2 M. a therefore DF— a .J x>
EF fcall be, as in the Prop. 7ytI

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%6o The tenth Book of

PROP. CIX.

A medial rectan-
gle B being taken
from a rational rect-
avgle A-+B, the right
line H which contain:
in power the [pace re*
viaining A> is one of
thofe two irrational
lines, viz. either a refidual line y or a Minor line.
Upon CD p make the re&angles CIm A-+-B.
a if ax.i.and rl— B. whence CE a-=- An-fiq. wherefore
b hyp. rtwJ becaufe CI b is fv. c therefore CK is p tl CD.
conjfr. but being £ I b is pv> d fhall FK be p 'm. CD. e
c zr. io. whence CKTX FK. f therefore CF isarefidual
d 2 j. io. line. Wherefore if CK be tl J CKq - FKq,
e i$. io. g then CF fhall be a firft refidual, b therefore^
f 74- Ic- CE (H) is a refidual line. But if CK "ex
g iJef£$. CKq - FKq. k then CF fliall be a fifth refi-
io. dual ; and confequently H (y/ CE) / fhall be a
h 92. to. Minor line. JWnch was to be dem.
k4.<fe/.8$.

PP. OP. ex.

9S* *Q> See the free. Scheme.

A rational reel angle B being taken away from
a medial reB angle A^ B9 other two irrational lines
are madey namely either a jirft medial refidual line,
a 3; ax. j. or a liiie making with a rational [pace the whole
b hyp. and /pace medial.

cofijtr. ^ Upon CD the propounded p make the reftan-
e 25.10. glesCI- A-B, andFI— B. a whence CErrA
d Mi 10. 5= Hq. Therefore becaufe CI b is^y, c fliall CK
c 13. ic. be t TEL CD. but becaufe FI b is fo, dthence FK
f 74- 10. f I *B- CD. e whence CK FK. f therefore CF
g2.ie/.85. is a refidual, £ and that a fecond. If CK "n.
5* 4/ CKq - FKq, h then H CE) is a firft me-

h 93. 10. dial reiiduah But if CK tl V CKq — FKq. Jb
k 5.<fe/.8$. then fhall CF be a fifth refidual ; and / confe-
10. quently H (V CE) fhall be a line making fxr
1$<5*iq. withp^ JTbicIj wat to be dem.

PROF*

10.
1

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EUCLIDEV Elem^fu $61

PROP. CXT.

See the fame Scheme*
A medial fpace B being taken away from a medial
Jpacc A-+- B, which is mcommenfurable to the whole
A B) the other two irrational lines are made% viz.
either a fecond medial rejidual line, or a line making
with a medial $ ace the whole Jjtace medial.

Upon CD f make the rectangles CI^= A -t- B, a 3. ax. 1,
and Fh B. a wherefore Cfc— ArrHg. Becaufe b 13. 10.
therefore CI is fxyy b thence CK is p ~cl CD, c hyf*
and in like manner FK p tl CD. Likewife be- d 10. ic.
caufe CI c Tl FI, d therefore CK tl FK. e e 74. ic.
wherefore CF is a refidual, / namely a third. Iff 2.<fe/,8f.
CK tl V CKq -FKq, g whence H (v/CE mall 10.
be a fecond medial refidual, but if CK tl *J g 94. 10.
CKq— FKq h then lhall CF be a fixth refidual. h6Jc/.8f.
k wherefore A fhall be a line making ^ with j/. i0.
flPfoVA wtf * *0 be dem% k 97. io.

PROP. CXII.
A ' ■ ■ A refidual line A is not

D E the fame with a bimedial
line.

Upon BC propounded p
make the reftangle CD"
Aq. Therefore feeing A a 98. to.
is a refidual, a BD mall be

s

a firft refidual, to which let DE be the line con b 74. 10.
ruent^or that may be adjoined, b wherefore BE, c i.def*8$.
)E, are p ~r_}-. c and BE tl BC. If you con- 10.
ceive A to be a binomial, then BD is a firft bin.
whofe names let be BF, FD \ and let BF be c d $7. ro.
FD. d therefore BF, FD are 0 TL ; and BF e e I.Af.48.
TL BC. therefore fince BC tl BE f lhall BE be 10.
TL BF. g and thence BE TX FE. h therefore f 12, 10.
FE is p. Likewife becaufe BE Tl DE, k lhall gcor.16.10
FE be TL DE. / wherefore ?D is a refidual, h fch. !!•
and To FD is p. but it was lhewn p, which are 10.
repugnant. Thereto re A is fa lily conceived to k 14. iO.

kt a uiuginial. Winch was to be dm* 1 74-

1 $ $ The

Digitized

?6z

Tht tenth Book of

Zfo names of the x\ irrational lines differing

one from another*

1. A Medial line.

2. A binomial line , of which there are lis

lpecies.

3. A firft bimedial line, V-.'

4. A fecond bimedial. \ % #fcif '

5. A Major line. " k^

6. A line containing in power a rational fu-

perficies, and a medial fuperficicfc,,

7. A line containing in power two medial

fuperficies.

8. A re he ual line;of which there are alfo 6 kinds,

9. A firft medial refidual line. - V ,

10. A fecond medial refidual line, ^

11. A Minor line. '^J

12. A line making with a rational fuperficies
the whole fuperficies medial.

13. A line making with a medial fuperficies
the whole fuperficies medial.

Being the differences of breadths do argue diffe-
rences of right lines, whofe fquares are allied to
fame rational I'm, and it is demonffrated in the pee.
rropofitions that the breadths w^ch arife from ep-
flying of the fquares of thefe ij lines do differ one
from another , it evidently follows that thefe 13 lines
do alfo differ one from another. ^ •

prop, cjafcg J*

H~ ~ The fquare of a

n n binomial BC (BD
DC) makes the breadth
EC a refidual line, whofe
names BJff CH9 are
commenfurable to the
names BD9 DO, of

ihi binomial lm% ani

* in

to

igitized b

y Google

EUCLIDE'x Elements. 26;

in the fame proportion {EH. BD :: CH. DC.) and

moreover the ujidual line EC which u nude, is of
the fame order with BC the binomial. \

Upon DC the lefs name a make the reftanglea co>:i6.(:,
DF== Aq = BE. whence BC. CD b :: FC.CE.b 14. 6.
therefore by divifion, BD DC :: FE. EC. And
whereas BD e cr DC, d thence Ft foil Ac- c
EC Take EG =EC, and make FG. GE :: EC. a 14. 5.
CH Then EH, and CH mall be the names of
the refidual EC, whereunto all is agreeable that
is propounded in the Theorem. For being that
by addition FE. GE (EC) -.: EH. CH. therefore e 12. y.

FH, EH e :: EH. CH tej^B^SStW*
wherefore fincc BD^ tl DC. & thence foil EH g % .
be -xl CH. b and FHq Tu EHq. Therefore be- h ic. to.
caufeFHq EHq :: FH. CH. h lhall FH be tl kror.20.rf.
CH, / and fo FC TL CH. Moreover CDg isp,l 16. 10.
and DF (Aq) g is ov. m theretore FC is p, TL m zi. 10.
CD. whence alio CH isp tl CD. » therefore n Ich. iz.
EH CH are i and T-. . as before, 0 theretore EC 10.
js a'refidual line, to which CH may be joined. o 74- 10.
Furthermore EH. CH/:: BD. DC. and lo by
inverfron EH. BD :: CH. DC. whence becaule
CH f tl DC, p lhall EH be tl ED. But fup- p re 10.
pofe BD XL. V BDq-DCq. a then lhall EH be q ij. 10.
TL VEHq— CHq. Alio if BD xl p propound- r 12 10.
ed, then ifiall EH be tl to the fame p. J that I 1.A/.48.
is' if BC be a firft binomial, t EC mall be a firft 10.
refidual. In like manner, if DC be to the tl t i.def.5^
propounded p. r then is CH TL to the fame p. » 10.
that is, if BC be a fecond binomial, x EC lhall u zdef.^b.
be a fecond refidual; and if this bea third binom.ro.
then that mail be a third refidual, gfr- But jtx
BD be TL. V BDn - DCq, y then mail EH be 10.
TL V EHq — CHq. therefore if BC be a 4th, y 15. 10.
Sth, or 6th binomial, EG lhall be likewife a 4th,
Sth or 6th refidual. /^ie/-' was to be dem.
¥ 1

R4

Digitized by Googl

264

The tenth Bool of

* %

prop. cxrv.

ZD

r

I

The fjuare of a ratio*
nal line A applied to a
refidual line B C (BD
— CD) makes the breadth
BE a binomial \ whofe
names BE, GE are com-
vienfurable to the names
BD,BCof the refidual
line BC, and in the fame
proportion, and moreover, the binomial line which
is made {BEy is of the fame order with the refidual
line (BC.)

a tor .16.6. a Make the re&angle DF m Aq. and $F. FE
b iz. 6. b :: EG. GF. whence for that DFrrAqrrCE, 0
therefore BD. BC :: BE. BF, therefore by con-
verfion of proportion BD. CI):: BE. FE::EG.
GF :: d BG. EG. but BD e Q- CD. / therefore
BG-r^ G£. therefore becaufe BGq. GEq£::

c 14.
d 19.

c hyp

6.

f 10. 10

g*».io.6.BG. GF. b fhall BG be tl GF. k and tb'BG
h iu. to. BF. moreover BD e is p, and the reftangle
k cor. 16. DF (Aq)* is p,. / therefore BE is f tx BD. m
xo. therefore alfo BG is p XL BD. » therefore BG,
1 ii.io. GE are p e wherefore BE is a binomial.
xa 12. io.Laftly, becaufe BD. CD:: BG. GE. andinverfe-
nfcb. 12. ly BD. BG :: CD. GE. and BD tl BG. p thence
10. fhall CD be U GE. therefore if CB be a firft
o 37. 10. reGdual, BE fhall be a firft binomial, &c, asio
P io. 10. the prec. therefore, c.

PROP,

L

1

Digitized by Google

EUCLIDEV Elements.

... A

PROP. CXV.

>

' - ■

1

-

If a $ace JB be contained under a refidual lint
JC (CE — JE) and. a binomial CBy wbofe names
CDy DB are commenfurable to the navies £E,JE, of
the refidual line, and in the favie proportion (CE.
JE :; CD. DB) then the right line F which contains
in power that $ace JB9 is irrational.

Let G be p. and make the reftangle CH —
Gq; a then fhall BH (HI- IB) be a relidual Jine,* j 1 j; t<\
and HI a tl CD b ~cl CE. a and BI^x DB a b hyp.
and HI.BI :: CD.DBi :: CE.EA. therefore by in-
verfipn HI. CE :: BI. EA. c therefore BH, AC :: c 19. j.
HL CE:: BI.EA. c therefore BH.AC :: HI. CExd iz. 10.
BI. EA. wherefore fince d HI TX CE, e thence e 10. ia
BH xl AC. /therefore the reftangle #C n.BA. f 1. 6. atu\
But HC (Gq) b is fag therefore BA (Fq) is fp : g fib. ix^
and confequently F is L Winch wa* to be dem. 10.

1 ' Coroll;
Hereby it appears that a rational fuperficies may
be contained under two irrational right lines.

PROP. CXVI.

-ff Of a viedial line JB
are produced infinite inch-
tional lines BE7 EF9 Sec.
whereof none is of the fame
kind with any of the pre-
cedent.

Let AC be propounded L and AD a redtangle

coo;

Digitized by Google

zQS The tenth Book of

contained under AC, AB. a therefore AD is Spm
a few. 38. Take BE ■= s/ AD. £ then BE is p, and the
10. fame with none of the former. For no fquare of
b ii. 10. any of the former being applied to p, makes the
breadth medial. Let the re&angle DE be finifh-
ed, a then DE ftiall be fr. and frconfequently EF
(*/ DE) lhall be p, and not the fame with any
of the former, for no fquare of the former be-
ing applied to p, makes the breadth BE. there-
fore,

PROP. CXVII.

#

r

/ Let it be required tofiew t1>at
in fquare figures BD. the diameter
JC is incommenfurable in length
to the fide AB.

* 47* *• m 1 XI for ACq. ABq a :: 2. 1 b ::
bcor.z4.8- B C not Q. Q. e therefore AC U

f 9*10* AB. Wbichwas to be dem. This Theorem was
of great note with the ancient Phiiofophers ;
fo tnat he that underftood it not was efteemed
by Plato undeferving the name of a Man, but
to be reckoned among Brutes,

■ •

The End of tU ttnth Book,

• ■

If

>

. >

- •

» 1

4 *

THE

Digitized by Google

THE ELEVENTH BOOK

of

EUCLIDE'j ELEMENTS,

Definitions.

Solid is that which hath
breadth, and thicknefs.

II. The term, or extreme of a folid
is a Superficies.

III. A right line
AB is perpendicular
to a Plane CD, when
it makes right angles
ABD, ABE, ABF,
with all the right
lines BD, BF, BF,
that touch it, and
are drawn in the
faid Plane,

IV. A Plane AB,
is perpendicular to ft
Plane CD, when the
right lines FG, HK,
drawn in one Plane
AB to the line of
common fe&ion of
the two Planes EB,
and making right an-
gles therewith , do
he other Plane

alfb make right angles with

The eleventh Book of

.V. The inclination
of a right line AB to a
Plane CD, . i$, when a

perpendicular AE is
drawn' from A the
higheft point of that
line AB to the plane
CD, and another line
EB drawn from the

I v \^ .
f j

A.

/
/

E

point E, which the perpendicular AE makes
in the Plane CD, to the end B of the faid
line AB which is in the fame Plane, whereby
the angle is acute ABE which is contained
under the infilling line AB, and the line drawn
in the plane EB.

' VI. The inclina-

tion of a Plane AB,
to a Plane CD, is an
acute angle FHG
contained under the
right lines EH, OH,
twhich being drawn
JX> in either of the Planes
AB,CD.to the fame point H of the 'common
feftion BE, make right angles FHB, GHB, with
the common ledhon BE.

VII. Planes are faid to be inclined to other
Planes in the fame manner, when the faid angles
of inclination are equal one to another.

VIII. Parallel Planes are thofe which beipg
prolonged never meet.

IX. Like folid figures are fuch as are contain-
ed under like Planes equal in number,

X. Equal and like folid figures are fufh as
are contained under like planes equal both in
multitude and magnitude.

XL A folid angle is the inclination of more
than two right lines which touch one another*
and are not in the fame fuperfjeies.

EUCLIDF/ Elematts:

Or this;

A folid angle is that which is contained under
more than two plane angles not being in the fame
fuperficies but confifting all at one point.

XII. A Pyiarhide is a lolid figuie comprehend-
ed under divers planes let upon one plane,
(which is the bafe of the Pyramide) and ga-
thered together to one point.

XIII. A Prifme i.i a folid figure contained
under planes, whereof the two oppofite are
equal, like, and parallel ; but the others are
parallelograms.

XIV. A Sphere is a folid figure made when
the diameter of a circle abiding unmoved, the
femicircle is turned round about, till it return
to the fame place from whence it began to be
moved.

CoroU.

Hence, all the rayes drawn from the center
to the fuperficies ©fa fphere, are equal amongft
themfelves.

XV. The Axis of a fphere, is that fixed right
line, about which the femicircle is moved.

XVI. The Center of a fphere, is the fame
point with that of the femicircle.

XVII. The Diameter of a fphere, is a right
line drawn thro' the center, and terminated on
either fide in the fuperficies of the fphere.

XVIII. A Cone is a figure made, when one
fide of a reftangled triangle (viz. one of thofe
that contain the right angie)remaining fjxedjthe
triahgle is turned round about till it return to
the place from whence it fir ft moved. And if the
fixed right line be equal to trie other which con-
tains the right angle, then the Cone is a reftan-
gled Cone : but if it be lefs, it is an obtu[e-angle4
Cone ; if greater an acute-angled Cone.

XIX. The Axis of a Cone is that fix'd line
abgut which the triangle is moved,

XX. The

The eleventh Book of

XX. The Bafe of a Cone is the circle, which •
is defcribed by the right line moved about.

XXI. A Cylinder is a figure made by the mo-
ving round of a right-angled parallelogram, one
of the fides thereof, (namely which contain the
light angle) abiding fixM, till the parallelo-
gram be turned about to the fame place, where,
it began to move.

XXII. The Axisof aCylinder is that quief cent
right line, about which the parallelogr.is turned.

XXIII. And the Bafes of a Cylinder are the
circles which are defcribed by the two oppofite
fides in their motion.

XXIV. Like Cones and Cylinders, are they^
both whofe Axes and Diameters of their Bafes
are proportional.

XXV. A Cube is a folid figure contained un-
der fix equal fquares.

XX VI. ATetraedron is a folid figure contain-
ed under four equal and equilateral triangles.

XXVII. An Odtaedron is a folid figure contain-
ed under eight equal and equilateral triangles.

XXVIII. A Dodecaedron is a folid figure con-
tained under twelve equal, equilateral and equi-
angular Pentagones.

XXIX. An lcofaedron is a folid figure contain-
ed under twenty equal and equilateral triangles.

XXX. A Parallelepipedon is a folid figure
contained under fix quadrilateral figures, whereof
thote which are oppofite are parallel.

XXXI. A folid figure is faid to be infcribed
in a folid figure, when all the angles of the figure
infcribed are comprehended either within the
angles, or in the fides, or in the planes of the
figure wherein it is infcribed.

XXXII. Likewife a folid figure is then faid
to be circumfcfibed about a folid figure, when
cither the angles, or fides, or planes of the Cir-
C«> ifcribed figure touch *U the angles of the fi-
Ruxe which it Contains!

PROP.

Digitized by Google

EUCLIDE'f Elements:

PROP. I. 0*

One part AC of a right line
cannot he in a plane fupaficies.
and another part CB clevatel
upward.
Produce AC in the plane di-
re&lytoF. If you conceive CB to be drawn
ftrait from AC,then two right lines AB, AF, have
one common fegment AC. a IWnch is impojfible. a l©.a*.n

prop. n.

If two right lines AB, CD;
cut one another, they are in
the fame plane : And every
x , ^ triangle DEB is in one and
A O the jame plane.

For imagine EFG, part of the triangle DEB,
to be in one plane, and the part FDGB to be in
another, then EF part of the right line ED is
in aplane, and the other part elevated upwards^
a Which is ahfurd. Therefore the triangle EDB a I
is in one and the fame plane ; and fo alfo are
the right lines ED, EB ; a wherefore the whole
lines AB, DC, are in one plane. Which wa*
tobedemonftrated.

> • » >

PROP. III. *

- If two planes AB, CD, cut
7 one the other, their common
1? feit:on EF is a right line.
g If EF the common fe&ion
be not a right line, a then
in the plane AB draw the right line EGF, a and a t.pojl.u
in the plane CD draw the right line EHF,
therefore two right lines EGF, EtiF, include a
fuperficies. b Which is abfurd. b14.ax.1i

PROP.

. ir;

Digitized by Google

27*

The eleventh Book of .

Jf a right line EF he at right
angles ereBed upon two lines
JB^CDj cutting one the other,
at the common feciion E ; it
fiall alfo be at right angles to
the flane ACED drawn by the
[aid lines.

Take EA, EC, EB, ED,
equal one to the other, and
join the right lines AC, CB, BD, AD,t draw
any right line GH thro' E, and join FA, #C,
FD, FB, FG, FH. Becaufe AE is a = EB, and
DE a zsz EC, and the angle AED b — CEB. c
therefore AD is CB, c and likewife AC —
• DB. d therefore AD is parallel to CB, d and
AC to DB. e wherefore the angle GAE — EBH,
and the angle AGE-rrEHB. But alfo AE/=rEB.
g therefore GE^nEH, g and AGrrBH. whence
by reafon of th<? right angles, by the hyp. and
fo equal, atE, /nhe bafes FA, FC, FB, FD, are
equal. Therefore the triangles ADF, tBC, are
equilateral one to another, k and thence tJie an-
gle DAF^BCF.Tberefbre in the triangles AG F,
FBH, the fides FG, FH / are equal ; and fo by
confequence the triangle FEG and FEH are
mutually equilateral, m therefore the angles
FEG, FEH are equal, and wfo right angles.
In like manner, FE makes right angles with all
o jufef.ir.the lines drawn thro'E in the plane ADBC, o
and is therefore perpendicular to the faid plane.

i confix.
b 15. 1.
c 4. r.

e 29. i.

f conftr.

g I.

h 4. 1.

k& r.

14.x-

in 8. r.

n xo.def.i

PROP;

.1

uigi

Google

! A

V

EUCLIDE'* Elements. '2.7$

PROP. V.

If a right line AB he ereBei
perpendicular to three right lines
JCy JDy touching one the other
at the common feftionjhofe three
lines are in the fame plane.

For AC, AD, a are in one a z. H,
plane FC ; a and AD,AE, are
in one plane BE. which if you conceive to be
feveral planes, then let their interfedtion h beb
the right line AG \ therefore becaufe BA by
the Hypoth. is perpendicular to the right lines
AC, AD. c and fo to the plane FC, d it is alfoc 4. it.
perpendicular to the right line AG. therefore d $Jef.iu
(fince a that AB is in the fame plane with AC,
AE) the angles BAG, BAE, are right angles,
and consequently equal, the part and the whole.
Which is abfurd.

PROP. VI.

If two right lines AB, DC, he
erected perpendicular to one and
the fame plane EF thofe right
lines JB, DC are parallel one
to the other.

Draw AD, whereuntd let
DGrrr AB be perpendicular in
the plane EF, and joinBD,BG,AG. Being in the
triangles BAD, ADG, the angles BAD, ADG a a hyp.
are right angles, and AB h — DG, and AD is b confiu
common, c therefore BDis= AG. whence in thee 4% U
triangles AGB, BGD, equilateral one to the o-
ther, the angle BAG is d — BDG ; of which d 8. U
being BAG is a right angle, BDG fhall be fo
alfo. but the angle GDC is fuppofed right,there-
fore the right line GD is perpendicular to the
three lines DA, DB, CD. e which are therefore e J. ir.
in the fame plane / wherein AB is. Wherefore f 2. 11.
fince ABand CD are in the fame plane, and the
internal angles BAD, CDA, are right angles, gg 28.1.
AB and CD lhall be parallels. Winch wai to he
iem> S PROP.

Digitized by GoogI

*74

The eleventh Book of
PROP. VII.

If there he two parallel rigfa
lives AB, CD, and any -points
£, F, be taken in both of them 9
the line EF which is joined at
thefe points, is in the fame
plane with theparailels JBCD.
Let the plane in which AB, CD are, be cut
by another plane at the pokits E, F. then if EF
is not in the plane ABCD, it mall not be the
common fe&ion. Therefore let EGF be the
a 3. ir. Conimon feclion ; which a then is a right line,
b 1 4. m.i. t}ierefore two right lines EF, EGF, include a

b Which is abfurd.

fuperfieies,

PROP. VIII.

IE

33

G

If there be two parallel right
lines AB, CD, whereof one
AB is perpendicular to a plate
EF. then the other CD JhalL
be perpendicular to the fame
j plane EF.

The preparation and de-
monftration of the fixth of this Book being,
transfev'd hither j the angles GDA, and GDS
are right angles : a therefore GD is perpendicu-
b 7. 11. lar to the plane, wherein are AD, DB (tii*
C3.^/.n.which alfo AB, CD are,) c therefore GD is*
perpendicular to CD. but the angle CDA is alfo
d r. 4 a* right angle, e therefore CD is perpendicular
* 4- H» to the plane EF. Winch was to- U dem.

»4. IT,

EUCLIDE'i Element*

PROP. IX.

*7?

IT B Kgbt lines (AB, CD) which

"VCr are parallel to the fame right

/ " line EF, hut not in the fame
CI D flane with 2"/, are alfo parallel

one to the other.
In the plane of the parallels AB, EF, draw
HG perpendicular to EF ; alfo in the plane of
the parallels EF, CD, draw IG perpendicular to
EF. therefore EG is perpendicular to the plane a 4* tt.
wherein HG, GI are ; and AH, CI are perpen- b 8. 11*
dicular to the fame plane, c therefore EH and c 6* 1+
CI are parallels. Winch was to he dem.

B

r

I

A

f ROP. X.

■ If two right lines AB, AC, touching
one another he parallel to two other right
C lines ED, DF, touching one another, and.
not being in the fame flane, thofe right
lines contain equal angles, B AC, EDF.
Let AB, AC, DE, DF, be equal
_ one to the other, and draw AB, BC,
1 EF, BE, CF. Being AB, DE, a are

fey

parallels, and equal, h alfo BE, AD, are paral- a W« ailA
lels and equal, In like manner CF, AD, are pz~conftr-
rallels and equal 5 c therefore alfo BE, FC, are b J?1 *•
parallels and equal, d Therefore BC, EF are e- c z: aK* r*
qual. Wherefore fince the triangles BAC, EDF, and I0, 1 ■
are of equal fides one to the other, the angles'* }}* u
BAC, EDF e fhall be equal* WhkhwastoUdcvu e &

PROP. XL

■M3

ft E * C

From a point given on high
A9 to draw a right line Al
erpendicular to a plane be-
ow BC.
In the plane BC draw any

Digitized by Google

276 * : The eleventh Book of

a 12. t. line DE ; to which from the point A a draw the
b 11. 1. perpendicular AF, and £ likewife FH in the

{)lane BC cutting thefaid line DE at F ; a then
et fall AI perpendicular to FH, Which AI
t Ihall be perpendicular to the plane BC.

c )1. 1. For thro'lc let KIL be drawn parallel to DE.
d conftr. Becaufe DE d is perpendicular to AF, and FH, e
e 4. 11. therefore DE fliallbe perpendicular to the plane
f 8. ii. IFA. and fo alfo KL / is perpendicular to the

J53Jg/l 11. fame plane, g therefore the angle KIA is a right
1 conftr. angle, but the angle AIF is alfo h a right angle.
1 4. 11. I therefore AI is perpendicular to the plane BC.
Whkh was to he done.

a 11. 11.

b ji. t.
c 8. EI.

a 6. n.

ZS5Z

PROP. XII.

m In a plane given BC, at a point
given therein A, to ereB ,a per-
fendicular line AF.
w From fome point without the
plane D, a draw DE perpendi-
cular to the faid plane BC, and joining the
points A, E, by a line AE, b draw AF parallel
to DE. c it is apparent that AF is perpendicu-
lar to the plane BC. Which was to be done.

This and the preceding Problem are prafli-
cally performed by applying two fquares to the
point given 5 as appears by 4. 11.

PROP. XIII.

3>!E

As a point given C in a
plane given AB, two right lines
CD ,C£L cannot be ereftedpev-
pendiadar on the fame fide.
_ For both CD, and CE, a
fliouid then be perpendicular to the plane AB,
and confequently parallels ; which is repugnant
to the definition of parallel lines.

*r6p>

uigmz<

ed by Google

EUCLIDEV Element's.

PROP. XIV.

277

Planes CD, FE , to winch the
J ame right line AB is perpendicur
lar, are parallel.

If you deny this ; then let
the planes CD, FE, meet, fo
that their common fe&ion be
the right line GH, in which
take any point I, draw to it

the right lines IA, IB, in the laid planes, where- y .
by in fhe triangle IAB, two angles IAB, IBA a a hh an*
are right angles, b Winch is abfurd. ' b 17 i

ibfi

PROP. XV.

If two right lines AB, AC,
touching one the other, be paral-
lel to two other right lines DE,
DF, touching one the other, and
not being in the fame plane w ith
them, the planes BAC, EDF9
drawn by thofe right lines are pa-
rallel one to the other.
From A a draw AG perpendicular to the r
plane EF. tandletGH, GI be parallel to DE,u „ '
DF. c thefe alfo fhall be parallel to AB, AC. c ]V \\
Therefore fince the angles IGA, HGA, d def'u.
light angles, alfo CAG, BAG, e fhall be right e £
angles, /therefore GA is perpendicular to thej J tI[
plane BC ; but the fame is perpendicular to the ac0nflr.
plane EF. /; therefore the planes BC, EF, arehi4. ii.
parallel. Which was to be don. r T * *

S J PROP.

w

278

% U II.

a t6\ 11.
b 1. 6.

Tie eleventh Book of
PROP, XVI,

B P

7/* /iro parallel planes AB^
CD, foa/* by fome other plane
HEIGF , f/;eir common fe-
ciions EHy GF are parallel
one to the other.

For if they be conceived
to be otherwife j being in
the fame plane that cuts
them, they will meet fome-
fuppofe in I ; wherefore

where, if produced ,

fince the whole lines HEI, FGI a are in the
planes AD, CD, being produced, the planes
alio fliall meet, contrary to the Hyp.

PROP. XVII.

If two right lines ALB, CMD, U
cut by parallel planes £F, GH> IK I
theyjball be cut proportionally. (AL.
LB:;CM.MD.

Let the right lines AC, BD, be
drawn in the planes EF, IK , as
alfo AD palling thro' the plane
GH in the point N. and join NL,
LM the planes of the ttiangles ADC, ADB,
make the feclions DB, LN, and AC, KM a
paiallels. Therefore AL. LB :: AN. t%
CA1. AID. Which wot to be dcm.

u

icf|

\

Q

DK

PROP,

, t Digitized by Google

EUCUDE'* Elements.

*79

PROP. XVIIL

>

If a right line AB be
perpendicular to fame
plane CD, all the planes
extended by that right
line JB (EFJkc.) Mat
he perpendicular to tfje
fame plane CD.

Let there be fome
plane EF drawn by AB, making the fe&ion EG a 3f •
with the plane CD ; froJJi fome point whereof b 8.ir.
H, a draw HI parallel to AB iai the plane EF \ c 4.^.11.
b then mall HI be perpendicular to the plane
CD, and fo likewife any other lines, that are
perpendicular to EG. b therefore the plane EF
is perpendicular to the plane CD ; and by the
lame reafon any other planes drawn by AB fhall
be perpendicular to EF. Which was to be dcm.

PROP. XIX.

• If two planes AB, CD,
cutting one the other, be
perpendicular to fome plane
GH. their line of common
fettion EF Jball be papen-
dicul.ir to the fame plane
(OH.)

Becaufe the planes AB, CD, are taken perpen-
Icular to the plane GH, it appears by4.def.1r. a 13.

I f* 1 m w \ 1 Ik •

that out of the point F there may be drawn iri
both planes AB, CD, a perpendicular to the
plane GH. which fhall be a but one ^ and theic-
ibre the common lection of the laid planes.

Wbifb was to be Am*

11.

PEOP.

Digitized by Google

2. 8b The eleventh Book of

PROP. XX.

If * foM Mgle ^BCD be con- /
VfV taincd under thee plane angles,

A^X BAD, DAC, BACy any two of

them howfoever' taken are greater
IB E C than the third.

If the three angles are equal, the affertion is
evident;if unequal,then let the greateft be BAC;
from whence a take away BAE ~ BAD, and
« r. make ADrrAE 5 and alfo draw BEC, BD, DC.

Becaufe the fide B A is common, and AD b—
b conp. AE ; and the angle BAE h = BAD. c thence is
c 4- *• BE == BD. but BD •+ DC is d cr BC. e therefore
d 2C. 1. DC cr EC. Wherefore fince AD b == AE, and the .
e 5 ax. r. fide AC is common, and DC c EC. / the angle
f z%. I. CAD fhall be cr EAC. # therefore the angle
g 4; 1. BAD -+ CAD cr BAC. Winch was to be dem,

PROP. XXL 1

Every folid angle A is cap-
tained under lefs angles than
four flane right angles.

For let a plane any-wife
cutting the fides pf the fo-
^ lid angle A make a inany-
C fided figure BCDE, and as
many triangles ABC, ACD, ADE, AEB. I de-
note all the angles of the polygone "by X > and
I term the fum of the angle at the bafes of the
*|1»S» & triangles Y. wherefore X -+ 4. right angles a =2
fch. 31. i, Y -h A. but being that, (of the angles at B) b
b 20. ir. the angle ABE •+ ABC is crCBE, and the fame
e 5- **• i.is true alfo of the angles at C, at D, and at E,
c it is manifeft that Y is c X. and confe-
quently A fhall be -3 4 right angles. Winch
rvas to be dem.

PROP.

/

Digitized by Google

EUCLlDE'x Element?,

4

PROP- XXIL

z9f

If there he three f lane angles A,ByHCI, whereof
two bowfoever taken are greater than the third, and,
the right lines which contain them he e$ualJD,JE,
FBy Sec. then of the right lines BE, FG, HI,
coupling thofe equal right lines together , it is jpojji-
lle to make a triangle.

A triangle may be a made of them, if any a 22. 1^
two be greater than the third : but they are fo.
For h make the angle HCK — B, and CK = b 2$. U
CH, and draw HK, IK c thence HK=:FG. and c 4. 1.
becaufe the angle KCI d cr A. e therefore Kir- d typ.
DE. but KI / t» HI -+ KH (FG.) therefore DE e 24. iJ
~3 HI -+ FG. By the like argument any two f 20, u
may be proved greater than the third ; and con-
fequently a it is poflible to make a triangle of
them. Winch was to he dem.

PROP.

Digitized by Google

*

liz . the tlevtnxh Book of

PROP. XXIII

«

« r

' To make a folid angle HHIK 0/ three-plane Angles
jf, 2?, C, whereof two bowfoever taken are greater
* 21* II* than the third. * But it is neceftary that thoje three
angles he lets than four rigJrt angles.

Make AD, AE, BE, BF, CF, CG, equal one
to the other ; and of the fubtended linesDE.EF,
a zi. 11. FG (that is, of the equal lines HI, 1K,KH) a
0nd zz. 1. make the triangle HKIj about which h defcribe
b 5. 4. tiie circle LHaI. * But becaufe AD is c HL.
*$eeCla-c let ADq be == HLq LMq. d and let LM
viv*. be perpendicular to the plane of the circle HKIf'
c/c^47.i.and draw HM, KM, IM. wherefore fince the
d ii. it, angle HLM* is aright angle, /thence is MHq
3.&/.11. — HLq LMq g =: ADq. therefore MH =
f 47- J. AD. By the fame reaforj MK, MI, AD (that is,
g conftr. AE, EB, &c.) are equal ; therefore fmce HM =
h conftr. AD, and MIrrAE, and DE = HI. k the angle
k 8. u A fliall be =HMI, fcaslikewife the angle IAlK
= B, k and the angle HMK = C. wherefore a
folid angle is made at M of the three given*
plane angles. Which was to be done. AD is a/Tu-
rned to be sr HL. But this is manifeft. For if
I conftr.^ AD be — or ~3 HL, then is the angle A / =
3. 1. m or c* JiLL In like manner lhall B be — or c
in zz. r. HLK, ami C = or cr KLI. wherefore A+B-+C
^4.^.15. * lhall either equal or exceed four right angles,
f . contrary to tJje Hypoth. therefoie rather let AD

be cr JiL. Which was to be dwu

PROP,

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CODE'; Element?,

PROP. XXIV.

7

)/

A

1

If a folid AB le con*
tained under parallel
planes, the off ofite planet
thereof ( AG, DB, &c. )
are like and tjual paral-
lelograms.

The plane AC cut-
ting the parallel plane?

V-4

AG, DB, a makes the feftions AH, DC, paral- a i& ir,
lels. and by the fame reafpn AD, HC are pa-
rallels. Therefore ADCH is a parallelogram.
By the like argument the other planes or the
parallelepipedon are b parallelograms wherefore b l^defi%
being AF is parallel to HG, and AD to HC, *c 10. n,
the angle FAD (hall be — CGH. therefore be-d ?4- *•
paufe AF d = HG, and AD d ^ HC, and fo e 7- 5*
AF. AD :: HG. HC, the triangles FAQ, QHC, g 6. 6.
g are like and h equal ; fqd confequentiy the h 4. i.
parallelograms AE, HB are like and k equal, and k 6. ax. ft
the fame may be lhewn of th$ otl^er bppof'

planes, therefore, &cr/

PROP.

D T C

' A A

0

' t

I H

if /I /b/M ParaU
E leleprpedon ABCD
be cut by a plane
EF9 Parallel to the
ofpojite planes ADy
BC ; then as the

bafe Btf, fo fbaH
folid AHD be to
folid BHC.

Conceive the Parallelepipedon to be extended
on either fide, and takeAb^AE, and BK-EB, a j&r.gf
and put the plane 1Q.,KP, parallel to the planes 1 Jef 6
AD, BC; then the pgrs.IM,AH,and^ DL,DG>* b 24.11."
and 1QL, AD, EF, are a like and equal, c c I0J fof
Tri^urciheParalle^ and lr, ' '

f

the eleventh Book of

by the fame reafoii the Parallelepipedon BP ^
BF. therefore the folids IF, EP are as multiple
of tkefolidsAF, EC, as the bafes IH, KH, are
of the bafes AH, BH. And if the bafis IH be

fl 24. it.cr, =, KH, d likewife fhall the folid IE be

and 9. z=, "3 EP. e confeauently AH. BH :: AF.

Hi. EC. Which was to be dem.

c 6t def. J. The fame may be accommodated to aU forts of
Pr'ifms, whence r

Coroll.

If any Prifm whatsoever be cut by a plane*
parallel to the oppofite planes, the feftion fhall
be a figure equal and like to the oppofite planes.

PROP. XXVI.

Upon a right line
given JBy and at a
point given in it J9
to make a folidangla
AH1L equal to a
folid angle given
CDEF.

£ ii% 11. From fome point F a draw FG perpendicular
to the plane DCE, and draw the right lines DF,
FE, EG, GD, CG. Make AH = CD, and the
angles HAI = DCE, and AI = CE ; and in the
plane HAI make the angle HAK == DCG, and
AK ee CG. then ered 1CL perpendicular to the
plane HAI, and let KL be rrGF. and draw AL:
then AHIL ihall be a folid angle equal to that
given CDEF. For the conftrudlion of this does
wholly refemble the framing of that, as may ea-

% appear to any that examine iu

/

PROP,

■A

H

A

EUCLIDF* Element}]

PROP- XXVII.

Ka 7iD Vfona nghtlint

given AB to defcribe
a farallelepipedon
AK, like, and in like
manner fituate, with
a J olid parallelepipc-
dongiven CD.

7ra 1

Of the plane angles BAH, HAI, BAI, which
are equal to FCE. EGG, FCG, a make the folid a 26. «;
angle A equal to the folid angle C; alfo b make b 12. 6\
FC. CE :: BA. AH. b and CE. CG :: AH. AI (cc zz. 5.
whence of equality FC.CG :: BA. AI) and finifli
the parallelepipedon AK, which mall be like to
that which is given*

For by the conftru&ion, the Parallelogram dd i.def.6i
BH is like FE, and d HI to EG, and d BI to FG,
and e fo the oppofites of thefe to the oppofites e 14. n*
of them : therefore the fix planes of the folid
AK are like to the fix planes of the folid CD,
/ and consequently AK, CD, are like folids,f 9Aef.11
Which was to he denu

PROP. XXVIII.

If a folid parallelepipedon
AB be cut by a plane FGCD
drawn by the diagonal lines
DF, CG, of the oppofite
planes AEy HB, that folid
AB Jhall be equally bife&ed
by the plane FGCD.
For becaufe DC, FG, are a equal and parallels,a
I the plane FGCD is a Pgr. and being a the Pgrs. b
AE, HB, are equal and like, b alfo the triangles
AFD,HGC,CGB,DF£ are equal and like. But
the Pgrs. AC, AG, are equal and like to FB and
FD, therefore all the planes of the prifme FGC-
DAH are equal and like to ail the planes of the
prifme FGCDEB, andcconfequently this prifme

24* ii.
34. 1.

is equal to that, UHjich was to be denu

Qfydefn*

PROP.

Digitized by

*8£

-3>

T&e eleventh Booh of
PROP. XXIX.

* 7. e. J*-

tween the
parallel
panes AG-
HE, FL-
KD,andfo
tmderftand
it in the
following.
a 10. def
ji.and 5 j
I*

b ft

So/id Varallelepipedons AGHEFBCD, AG HEM-
LKI9 being conjfituted upon the fame bafe AG HE
and * in the fame height % whofe infifling lines Ap\
AM> aye placed in the fame right lines AG, FLy are
equal one to the other.

For a if from the eqiial prifmes AFMEDI,
GBLHCK, the common prifme NBMPCI be
taken away, and the folid AGNEHR be added,
the Parallelepipedon AGHEFBCD fliall be ^
AGHEMLKI. Which was & be dem.

PROP, XXX.

Solid paraUelepipedons ADBCHEFG, ADCB1M
LKkeing conflicted uponthe fame bafe ADBC, and

Digitized by Googl

EUCIIDFi EUmentu **%

in the fame height, whofe infijling lines JB* At
are ntt Placed in the fame right lines, are equal
one to the other.

For produce the right lines HEO, GFN, and
LMO,KIP;aaddrawAP,DO,BQ,CN. Athena 34. 1.
Stall DC, AB, HG, EF, PQ, ON be as well e-
cual and parallel one to the other as AD, HE,
GF BC, KL, IM, QN,PO. h wherefore the pa- b 29. ft*
rallelepipedon ADCBPONQ. mall be equal to
either parallelepipedon ADCBHEFG, ADCBI-
MLK 5 and c conlequently thefe two are equal c *• *•
one to the other. Winch was to be dem.

PROP. XXXI.

■

* ly height

S under ft and
Sohd parallelepipedon, ALEKGMBI, CPaOH^ tbeperpen-
D N, being conftituted upon equaibafes ALEKfiPeaO* dicular
/tnd*in the fame height are equal, one to the other, drawnfiom
Firflr, let the parallelepipedons AB, CD, have theplaneof
the fides perpendicular to the bales, and at the the bafe to
fide CP being produced, a make the parallelo- the opp9ftto
gram PRTS equal and like to the parallelogram plane.
KELA. b and f© the parallelepipedon PRTSQ: a 18. 6.
VYX equal and like toihe parallelepipedon AB. b 27. it.
Produce 0«E, ND<T, «PZDQF, ERB, fVy, andioJef.
TSZ, YXF, and draw E JS By, ZF.

The planes Os/N, CRVHZTYF c are paral- c 50. def.
lels one to the other , d and the pgrs. ALEK, 11.
CD«0, PRTS, PRBZ are equal. Therefore fince d hyp. ani
the parallelepipedon CD.PV :: pgr.C*(PRBZ) ? 5.
9 :; parallelepipedon PRBZQVyF.PVc^^the e 11.

paral-

1

f

* 9

88

29. ir.

conftr.

The eleventh Book of

parallelepipedon CD/lhall berrPRBZVCKF^=
PRVQSTYX /; = AB. Which was to be dem.

But if the paralielepipedons AB, CD, have
fides oblique to the bafes, then on the fame ba-
tes andinthe fame heighth place parallelepipe-
k 29. ir. dons whole fides are perpendicular to the bafe.

k They fliall be equal to one another, and thofe
m ikrtx.i. that are oblique ; m' whence alio the oblique
paralielepipedons AB, CD are equal. IVTiich
was to be dew*

• *

PROP. XXXIL

1

Solid paraUelepipedoris ABCD, £FGL, of the fatiit
teigbtb, are one to the other, as their bafesJB^EF*
a 45. 1. Produce EHI, a and make the pgr. FI = AB,
b gr. 1. and * compleat the parallelepipedon FINM. It
c 31. 11. is clear that the parallelepipedon FINM. U
d25.11. ABCD.) EFGL d :: FI (AB.) EF. Winch was
. to be dm.

k 3* 1*

PROP. XXXIII.

Like folid parallelepiped
dons , JBCD EFGH, are
in tripled proportion one to
the other of that in which
their homologous /ides or of
like proportion Al, EK,
arc.

Produce the right lines
AIL, DIO, BIN, and a
make 1L,I0, IN, equal

to

Digitized by Google

EUCUDE'i Elements'. 489

to EX, KH, KF, h and fo the parallelepipedon b 27. it*
IXMT equal and like to the parallelepipedon c 31. 1.
EFGH. c Let the paralleps.LXPB, DLYQ_ be fi- d hyp.
nifhed. d Then (hall be AL. IL (EK) :: DI. IO e 1. 6.
(HK):: RI. IN- KF. e that is, the Pgr. AD. DL f j*. jU
:: DL. IX :: BO. IT. f i. e. the parallepp. A BCD. g conftr*
DLQY::DLQY.IXBP .IXBP.IXMT.feEFGHj fi\o.^/.j.
h therefore the proportion of ABCD to EFGHk 1,6%
is triple of the proportion of ABCD to DLQY,
k or of AI to EK. Which was to he denu

Cor oil.

• ■

Hence it appears, that if four right lines be
continually proportional, as the firft is to the
fourth, fo is a parallelepipedon defcribed on the
fiift to a parallelepipedon defcribed on the f«-
cond, being like and in like manner defcribed,

PROP. XXXIV.

5p In equal folid pa-
'TFraUelepipedons JDCB,
y EHGF, the hafes and
altitudes are recipYO-
T cal (AD. EH :: EG.
a AC.) And [olid pa-
rallelepipedo7is,JDCB9
EHGF, whofe hafes and altitudes are reciprocal,
ere equ.U.

Firft, let the fides CB,GE be perpendicular to a J< *<
thebafes ; then if the altitudes of thefolids are b 31, 1.
equal, the bafcsalfofllall be equal, and the thing c \u it.
is clear. But if the attitudes are unequal, from d 17. $2
the greater EG a take EI ^AC, and at I b draw et.6. J
the plane IK parallel to the bafe EH. then f conjlr.

1. Hyp. AD. EH c :: parallepp. ADCB.EHIK gin j.
<J:: parallepp. EHGF. EHIK c :: GL. IL e :: GE* h 32. 11J
IE. ff AC.) £ it is plain therefore that AD. EH k hyp.

GE. AC. Winch was to be dem. 1 U 6*

2. Hvp. ADCB.EH1K b:: AD.EHfc::EG.EI/m $2. if,
GL, IL m ;: parallepp. EHGF. EHIK* ?t where- n 9. j*

tyre

Digitized by Google

7&e eleventh Book of

fore the parallelepipedon ADCB jfc EHGF.

Which was to be dem.

Moreover, let the fides be oblique to the bafes^
and eteft right parajlelepipedons upon the fame
bafes in the fame altitude } the oblique paralle-
lepipedons fhall be equal to them. Wherefore
fince by the firft part, the bafes and altitudes
of thofe be reciprocal, the bafes andaltitudes
of thefe alfo fliall be reciprocal. Which was to
ie dewm

Coroll:

JU that hath been demonjtrated of parallelepipe-
don* in the 29,30,51,31, 3 3,34 Prop, does alfo agree
in triangular prif me s, which are half parallelepiped
donsy as appears by Prop. 28. Therefore,

1. Triangular prifmes are of equal heighth
With their bafes.

2. If they have the fame or equal bafes and
the fame altitude, they are equal.

3. If they be like, their proportion is triple
to that of their fides of like proportion.

4. If they be equal, their bafes and altitudes
are reciprocal ; and if their bafes and altitudes
be reciprocal, they are alfo equal.

prop; xxxv.

If there be two
plane angles BAC,
EDF, equal, and
from the points of
thofe angles two
right lines AG, DH
be eletated on high,
contain'irg equal angles with the lines firft given ^each
toMs correjfiondent angle (the angle GAB — HT)E%
and GACzzHDF.) and if in thofe elevated lints AG,
i)H,fome points betaken.Gyff'-, and font thefe points
fetfendicubr lives GtHK, drawn to theplanes BACf
£DF7 in which the angles firft given are, and n<< H

urn

Digitized by Google

EUCLIDE** Eiementi. 29I

lines AI% DK, he drawn to the angles firft given from
the points I K> which vre made hy the perpendiculars
in the planes, thofe right knes with i be equated lines
JG, DH pall contain equal anglesGAM, HDK.

Make DH, AL, equal ; and GI,LM parallels,
and MC 10 AC, MB to AB, KF to DF, Kfi to
Dk perpendicular; and draw,the ngiu luie> EC,
£8, LC, and EF, HF, HE \ a and LM is per-ag. ir;
pendicular to the plane BAC 5 h wiieiefore iheD ^defiu
angles LMC, LMA, LMB \ and by the lame
leafon the angles HKF, HKD, HKE are right
angles. Therefore ALq c LMq i " AMqc--c 47.1.
LMq-CMq \ ACq c LCq - ACq. d there- 4 46. i<
fore the angle ACL is a right angle. Again e 47,
ALq e - LMq MAq e =c LMq + *Mq
BAq e -r= BLq BAq. d therefore the angle
ABL is alfo a right angle. By the like infe-
rence the angles DFH. DEH are right angles ; /
therefore AB=rDE, / and BL -EH, /and ACf 26. ^
r=DF, and CL— FH. # wherefore alfo BC — g4. u
£F; g and the angle ABC = DEF, g and the*
angle ACB = DFE. whence the other right h t. ax.u
angles CBM, BCM, aie equal to the other FEK, k 26. 1.
EFK. k therefore CM r= FK, / and fo alfo AMI 47.
r=z DK. therefore if from LAq m — HDq be m f07J^r< ,
taken away AMq 2SS DKq, n there remains 1147.1.8?
LMq = HKq. wherefore the triangles LAM, j.^J -
HDK are equilateral one to the other } 0 0 g. I.
therefore the angle LAM = HDK. Which
wos to he dem.

Coroll.

Therefore, if there be two plane angles equal,
from whofe points equal right line^ be elevated
on hifch, containing equal angles with the lints
firft giwn, each to each j perpendicular^ didwrt
from the extreme points of thole elevated lines
to the plants of the angles firft given, are equal
©ne to the ctlvi , viz. LM = JfelK.

1.

PROP!

Digitized by Google

1^2

Tie eleventh Book of

r •

PROP. XXXVI.
XT j. i .j.>t If theft he three
n ^ right lines DE,DG>
DF proportional^ the
J olid pat allelepipedon
DH made of them,
is equal to the folid
parallelepipedon IN
made of the middle DG(IL)rvhicb is alfo equilateral,
and equiangular to the [aid parallelepipedon DH.
a hyp* Becaufe DE. IK a:: IL. DF. b the parailelo-
b 14.6. gram LK lhall be r= FE. and by reafon o£ the
equality ot the plane angles at E and 1, and of
the lines GD,IM, alfo the altitudes of the pa-
C31.il. rallelepipedons are equal by the preceding Co-
rollary, therefore the parallelepipedons are equal
one to the other. Wliich was to be dem.

PROP. XXXVII.

1

ZE7

Y.

£71
&0

If there be four right lines A,B,C,D, proportional,
the folid parallelepipedons A,B£yD being like, and in
like fort defcribed from themfb all be proportional. And
if the folid parallelepipedons, being tike and in like
fort defcribed, be proportional (A. B :: C. D) then
thofe right lines A,B,C,D, JbalL be proportional.
For rhe proportions of the parallelepipedons
a ir. ti are triple of thofe of the lines ; therefore if A.
b/r/;.i?.$.B :: C. D. b then lhall the parallelepipedon A.

parallelepipedon B :: parallelepipedon C. paral-
. lelepipedon D. and foalfo comranly.

PRO?.

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EUCLIDE'r Elements.

I

PROP. XXXVIII.

J/ a plane AB U
perpendicular to a

• i/'i . *

.*

p <^ p plane AC\ and a per*

pendicular line E F

be drawn from a point

E in one of the planes (AB) to the other plane ACf
th.it perpendicular EFfiaU fall upon tbecontm§n fe~
Bion of the f lanes AD.

If it be poifiolc, let F fall without the inter-
feron AD, and in the } lane AC a draw FG a iz* I.
perpendicular to AD, and join £Q. The angle
IGE b is a right angle, and EFG is fuppofed to b + and j,
be i'uch alfo \ therefore two right angles are in def. II.
the triangle EFG. c Which is abfurd. c 17. 1.

^ PROP. XXXIX. \!

E If the fides (AE, FCf

AF, EC\ andbH, GBf
DG, HB) of the oppofite
planes ACyDB, of a lolid
paratteicf ipedonABi he di-
vided into two equal
parts and planes ILt
PKMRy be drawn
their /e£lionsythe common
fettion of the planes ST9
and the diameter of tht
fohd paratielepipedon AB Jball divide one the other
into two equai parts. ' . r k 1" /

Draw the right lines SA,SC,TDJB . Becaufeb z9;«.
a the fides DO.OT are equal to the fides B^Ql-c 4- t.
J IS the alternate angles TOD TOJ equal al-d/,i.fj,t.
fo, c the bate e DT, TB, and the angles DTO.e h- *•
BTQ are equal, d therefore DTB is a right line,f 9. 1 1&
and fo in tfke manner is ASC. Moreover e mi.**.
well AD is parallel and equal to FQ<9S FG to g ». I.
CB, and / thence AD is pa«U«l e^ak to 4 h 7. 1 1,

X J * ■

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2^4 dwtntb Booh of

g $}•'!• g and cpnfequently AC to DB. b wherefore AB
h 7. ir. and ST are m the fame plane ABC D. Therefore

fince the vertical angles AVS, BVT,and the al-
k 7. ax. i. ternate angles ASV, BTV are equal ; k and AS
I *& t • *= BT ; therefore fliall AV be — BV. / and SV

= VT. Which was to he dem.

CorolL

Hence in every parallelepipedon all the diaflW-
texs bifcit one another in one point, V%

PROP. XL.

»• <

r 3b

7

B G

v

If two Vnfms ABCFED, GHML1K, he of equal
altitude, wbcteof one hath Us bafe ABCF a par a lie.
logram, and (be other GHM> a triangle } and if the
parallelogram A*CF be double to the mangle GHM\
thefe Vnfms A£CFEDr GHMLlKare equal
For if the parailelepipedons AN,G(^,bei

A n lhaii be equaL IVlnch was to be dem.

Andr From the preceding demonfi rations^ the dimenjion

Taca of triangular Pnfms, and quadrangular, or paralle-
*% lepipedom, is learnt \ viz. by multiplying the alti~
tude into the bafe.

r As if the altitude be i© feet, and the bafe loo
fquare feet (the bafe may be meafured by fch.%$.
i. or \yj 4it i.) thea multiply ioo by xof *d

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EUCLIDE'j Elements:

iooo cubic feet fhall be produced* for the fojidi-
ty of th* prifm give** *; I J

For as a redtangle, lb alfo is a right paralle-
lepipedon produced of the altitude multiplied
into the bale. Therefore every parallelepipedon
is produced of the altitude multiplied into the
bafe, as appears by 31. of this Book-
Moreover, fince the whole parallelepipedon is
produced of the altitude drawn into the bafe,the
half thereof (that is, a triangular Prifm) fhall
be produced of the altitude drawn into half the
bafe, namely the triple.

Up erve* That of tfajt -Utters which ^ejiote a [0-
lid angle, the firjl is 'always at the point in which

A } and the fupreme point of the Pyramide
BCDA is at the point A. and the .bafe is the
triangleBCD. 1 *

The End of the eleventh Book.

( - .* '

?4 ' THI

a?6

THE TWELFTH BOOK

OF

E UCLIDE'; ELEMENTS.

Ike nofxx onoxis figures ABCD$%FGWK>
deferred in circles ABD, FGI, die one
to another, as the fquares defcribed of
the diameters of the circles, AL, PM.
Draw AC. BL, FH»QM. Becaufe

P J if J. *>W • a*e ngflt w equal ; e therefore the
e 3*, 5. triangles ABL, FGM are equiangular. / where*
f w.4. 6 tore AB. FG .: AL. FM. * theiefoie ABCDE,
$zzX FGHIK :: ALq.FMcj. *

CoroJL

Henc- (becaufe AB. FG:: AL.FM ::BC.GH,

or ^ tlie ccmlems polvgonous figures de-

ft r.T2.gf fenced in a circJe are in h proportion as the
diameters,

PROP.

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EUCLIDE'x Elements.

297

m. f

PROP. H.

Circles JBT,EFNyarc in propor-
tion one to another, as the /quarts
0/ then diameters JC, EG are.

Suppofe ACq. EGq :: the cir-
cle ABT.I. I fay then I is equal
to the circle EFN.

For firft, if it be poffible, let
I be lefs than the circle EFN, and let K be the
excefs or difference. Infcribe the fquare EFGH
in the circle EFN, 4 it being the half of a cir- a /cb.7. 4*
cumfcribed fquare, and fo greater than the lemi-
circle. b Divide equally in two the arches EF, b jo. 3,
FG, GH, HE, and at the points of the divifions
join the right lines EL,LF, &c. at L draw the
tangent PQ.(V which isparallel to EF)and pro-c/cb.17. ?•
duce HEP, GFQ. then is the triangle ELF dd 41. 1.
the half of the parallelogram EPQF,andfo greater
than the half of the fegment ELF; and 111 like
fort the reft of thofe triangles exceed the halts
of the reft of the fegment.*. And if the arches
EL,LF,FM,£fr. be again bifefted, and the right
lines joined, the triangles "will likewife exceed
the half of the Tegmenta. Wherefore if the fquare
EFGH be taken from the circle EFN, and the
triangles from the other fegments, and this be
done continually, at length e there will remain e 1. 1$.
fome magnitude lefs thanK. Let us have gone fo
far, namely to the fegments EL, LF, FM, &c.

tajten

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±$$ The twelfth Bool of

f hyp. and taken together lefs than K. Therefore I f/the
a. ax. circle FENK) p the polyg. ELFMNHO (the
g jo.?, & circle FEN — the fegm. EL - LF, gfr.) In the
T. toft, i- circle ABT^ conceive a like polygonou AKBS-
h r. 12. CTDV infcribed. therefore firice AKBSCTDV.
k hyp. ELFMGNHO h :: ACq. EGq k :: the circle
1 9. ax. u ABT. I. and the polyg. AKBSCTDV / ~3 the
m 14. 5- circle ABT. the polyg. ELFMGNHO m fhali
be I. but before, 1 was -3 ELFMGNHO,
which is repugnant.

Again, if it be poflible, let I be cr the circle
n fafe EFN- Therefore becaufe ACq.EGq «:: the circle
ABT. I j and'in verfely L the circle ABT. EGq.
ACq. fuppofe I. the circle ABT :: the circle
o 14. 5. EFN. K. 0 therefore the circle ABTc K. p and
11. 5. E&q. ACq :: the circle EFN. K. which is fWwn
to be repugnant.

Therefore it muft be concluded, that 1 is zzz
to the circle EFN. Which was to be dem.

CorolL

Hence it follows, that as a circle is to a cir-
cle, fo is a po;ygonon defcribed in one to a like
polygonou defcribed in the other.

1 • ! »

. ; »« t •

f

PROP. III.

Every Pyt amide A B DC
having a triangular bafejnay
be divided ivto two Pyramid**
JEGH, HIKC, equal, and
like one to the other , havirg
bafes triangular, and like to
the whole ABDC ; and into
F X> two tfual Prifms, BFGEIH,
EGDHlKi which two Prifpis are greater than the
half of the whole Pyramids ABDC.

Divide the fides of the pyramide into two
parts at the pointsE,F,G,H,I,K,a«d join the right
linesEF,FG,GE,EI,IF,FK,KG,GH,HE.Becaufe
tbjt fides of the pyramide axe proportionally

cufc

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»

EUCLIDFx Elements:

cut, a thence HI,AB j and GF, AB -> and IFJJC^a 2. 6.
and HG, DC,^^. are parallels, and confequently
HI FG 5 and GH,FI are alfo parallels, therefore
it is apparent that the triangles ABD, AEG,EBF,
FDG,HiK, b are equiangular, and that the four b 19.1,
laft are c equal : in like manner the triangles c z6. u
ACB, AH E,EIB,HIC,FGK are equiangular ; and
the four laft are equal one to the other. Alfo the
triangles BFI, FDK, IKC, EGH 5 and laftly, the
triangles AHG, GDK, HJCC, EFI are like and
equal. Moreover the triangles HlK to ADB,and
EGH to BDC, and EFI to ADC, and FGK to
ABC, </are parallel. From whence it evidently d 15. 11.
follows, firft that the pyramides AEGH,HIKC
are equal, and e like to the whole ABDC,and to e 10. def
one another. Next, that the folids BFGEIH, n.
FGDIHX, are prifms, and that of equal heighth,
as being placed between the parallel planes ABD,
HlK, but the bale BFGE is /double of the bale f 2. ax. 1
FDG, wherefore the faid prifms are equal \ g 40. 1 1%
whereof the one BFGEIH is greater than the
pyramide BFEI, that is, then AEGH, the whoke
than its part , and confequently the two prifms
are greater than the two pyramides, and lb ex-
ceed the half of the whole pyramide ABDC.
Which was to be dem.

PROP. IV.

*'

If there be two jyramides ABCD, EFGH, of the
fame altitude, having triangular bafts J£C, UFG,

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3 00 The twelfth Book of

and either of them be divided into two pyr amides
(AILM, MN01) 5 and EPRS, STVH) equal one to #
the Qther. and like to the whole, and into two equal
prifms (IBKLMN, KLCNMO \ and PFJ^RST,
QRGTSV \) and if in like manner either of thofe
pyt amides made by the former divijion be divided^ and 1
this be done continually ; then as the bafe ofonepy-
r amide is to the bafe of the other pyramide, fo are all
the pifms which are in one pyr amide , to all the prifms
which are in the other pyramide, being equal in
multitude.

a IS* 5- For (applying the conftruftion of the prece-
b H.6. dent Prop.) BC.KC a :: FG.QG. £ therefore the
•triangle ABC is to the like triangle LKC as
d 16. 5. efq is to c the like RQG. therefore by permu-
tfeb. 34. tation ABCLFG d :: LKC . RQG e :: the prilm
*i- KLCNMO. QRGTSV ( for thefe are of equal
* 7- *• altitude) / :: 1KKLMN.. PFQRST. £ wherefore
g *• the triangle ABC. EFG ":the prifm KLCMNO
-*-lBK LMN- the prifm QRGTSV PFQRST.
Which was to be dem.

But if the pyramides MNOD, AILM \ and
EPRS,STVH, be further divided, in like man-
ner the four new prifms made hereby fhall be to
the four produced before, as the bafes MNO
and AIL are to the bafes STV, and EPR ; that
15, as LKC to RQG, or as ABC to bFG. h
wherefore all the prifms of the p} ramide ABCD.
are to all the prifms of the py ramide IFGH as
the bafe ABC is to the bafe LFG.

, ' • Digitized by Google

EUCLIDE'i Elements.
PROP. V.

K C F Q. GJ

Pyramids JBCDy EFGH, being ^nder the fame '
altitude, having triangular bafes JBCy EFGy are
one to another as their bafes ABC, EFG, arc.

Let the triangles AfiC. EFG :: A BCD. X. I
fay X is equal to the pyramide EFGH. For if it
be poflible, let X be "5 EFGH. and let the ex- .'7 -
cefs be Y, divide the pyramide EFGH into . j ,
prifms and pyramides. and the other pyramides
in like manner, a till the pyramides left EPRS, ' : ■
SVTH, be lefs than the folid Y. Therefore fincea *< Ie*
the pyramide EFGH = X -i Y, it is manifeft
that ths remaining prifms PFQRST, QRGTSV
are greater than the folid X. Conceive the pyra-
mide ABCD divided after the fame manner ; b bd, ft,
then will be the prifm IBKLMN- KLCNMO.
PFQRST QRGTSV : : ABC. EFG c :: the c hyp. .
pyr. ABCD. X. d therefore X c the prifm I'FQ-d 14. $•
RST -+■ QRGTSV ; which is contrary to that
which was affirmed before.

Again, conceive X c the pyr. EFGH. and
make the pyr. EFGH. Y :: X the pyr. ABCD ce hyp. aid
:: EFG. ABC. Becaufe EFGH/-3 X, g thence Y cor. 4. %%
"3 the pyr. ABCD. which isfhewn before to bttfuppof.
impoffible. Therefore I conclude, that X is equal g 14. 5^
to the pyr. EFGH. Which was to be dem.

PROP*

r

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302

c ii« 5*

1

d 5. 12.

f

PROP. VI.

Fyramides JBCDEF, GHIKLMyconfiftingunder
the fame altitude, and having folygonovs hates
JBCDE, GHIKLi are to one another as their ba-
fes JBCDE, FGHIKL are.

Draw the right lines AC,AD,Gl,GK. then is
the bafe ABC.ACD a :: the pyr, ABCF.ACDF*
I therefore by compofition ABCD. ACD :: the
pyr. ABCDF. ACDF. a but alfo ACD. ADE ::
the pyr. ACDF. ADEF. c therefore of equality
ABCD. ADE :: ABCDF. ADEF. and h thence
by compofition ABCDE. ADE :: the pyr. ABC-
DEF. ADEF. moreover ADE. GKL d :: the pyr.
ADEF. GKLM } and as before, and inveri'ely
GKL.GHIKL :: the pyr. GKLM. GHIKLM.
c therefore again ofequalityAbCDE.GHIKL::the
pyr.ABCDEF. GHIKLM. Wlnchwastobedem.

~wrk . If the bafes have not

™\ fides of equal multi-
tude, the demonftra-
tionwill proceed thus.
The bafe ABC. GHI c
::the pyi.ABCF, QH-
IK. e and ACD. GHI
C DH I .vthcpyr.ACDF.GH-
IK. / therefore the bafe ABCD. GHI the pyr.
ABCDF.GH1K. e Moreover the bate ADE.GHl
the pyr. ADEF. GHiK. f therefore the bafe
ABCDt, GHI the pyr. ABCDEF. GHIK.

PROP.

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EUCLIDE'i Elements:
PROP. VII.

Every Prifm, ABCDEF,
having a triangular bafef
may be divided into three
Pyr amides ACBF, ACDF,
CDFE, equal one to the
other, and having triangular bafes.

Draw the diameters of the parallelograms,
AC, CF, FD. Then the triangle ACB is a = a J4. r.
ACD. b therefore the pyramides of equal heighthb 5. 11.
ACBF. ACDF. are equal, in like manner the
pyr. DFAC = the pyr.DFEC. but ACDF and
DFAC are one and the fame pyramide. c there-c i.ax.i,
fore the three pyrfmides ACBF, ACDF,DFEC,
into which the Prifm is divided, are equal one
to the other* Which was to be dem.

Hence, every pyramide is
the third part of the Prifm
thai has the fa^ue bafe and
heighth with it j or every
prilm is treble of the pyra-
mide that lias the lame Dafe
and heighth with it.

For rtiol ve the polygonous
X prilm AECDEGHiKF into
J3 triangular Pii/ms ; and the
pyramide ABCDEH into triangular pyramides^
a then all the parts of the prifm fhall be treble a
to all the parts of the pyramide, b confequent- b i, $%
ly the whole prifm AfcCDEGHlKF is tre-
ble to the whole pyramide AfcCDEH. Which
was to be dem*

7. it.

1 •

frROP.

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504

Tie twelfth Book of
PROP. VIII.

£>

\

f

A C

Like Pyramides ABCD, EFGH, which have trian-
gular bates ABC> EFG are in triple proportion of that
in which their fides of like proportion AC, £G, are.
' * tj. tt« a Compleat the parallelepipedons ABICDM-
b9.def.iT.KL, EFNGHQOP, which h are like, and c
c 28. it. fextuple of the pyramides ABCD, EFGH. d and
and 7.12. therefore in the fame proportion with them one
d IS- 5- to another, e that is, triple of that of the (ides
« ??• II« of like proportion, fife.

CovolL

1

Hence, alfo like polygor.ous pyramides have
proportion tripled to that of the fides of like
proportion } as may eafily be proved by iefol-
ving the lame into triangular pyramides.

PROP, IX. v

* See the preceding Scheme.

In equal pyramides ABCD, EFGH, having trian-
gular Fafes ABC> EFG, the bajes and altitudes are
reciprocal ; And pyramides having triangular hafes,
whofe altitudes and bafes are reciprocal, are equal. ,
iJfyp.Tht compleated parallelepipedons ABIC-
a i8.li. DMKL, EFNGHQOP are a fextuple of the equal
*nd 7. 1*. pyramides ABCD,EFGH (either of either) and fo

equal

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EUCLIDE'f Ehwenti. $6j

equal one to the other, therefore the altitude

(H.) the altitude (D) b :: ABIC. EFNG r.vb^.u.

ABC. EFG. Winch was to be dem. q 15, 5.

i.Hyp. The altitude (K.) the altitude (D) d :: d hyp.
ABC. EFG e:: ABIC. EfNG. /thcretbie thee 15. j.
jparallelepipedons ABICDMKL, EFNGHQpPf 34. ir.
are equal, g confequently alfo the pyramides g 6.ax.t.
ABCD, EFGH being fubfcxmple of the fame,
are equal Winch was to be dem.

The fame is applicable to polygonous pyramides j
for they may alfo in like manner be reduced to tru
angulars.

Corott.

: Whatsoever u demonflrated of pyrtmides in Prop.6.
8,9. doeslikewife agree to any fort of pr if vis ; feeing
they are trif le of the pyramides that have the fame
lafe and altitude with them. Therefore

1. The proportion or prifms of equal altitude
is the fame with that or their bafes.
, 2. The proportion of like prifms is triple
of that of the fides of like proportion.

3. Equal prifms have their bafes and altitudes * „
reciprocal j and prifms which are fo reciprocal,
*re equal.

■

Schol.

m

* * 4 .* . • * * ■ *

From what is hitherto demonftrared the di-
menfion of any prifms and pyramides may be
collefted.

a The folidity of a prifm is produced of the a r.wi
altitude multiplied into /he bale * b and there- &fch. 40.
fore likewife that of a pyramide, of the third 11,
part of the altitude multiplied into the bafe, b 7. ix.

V

PROP.

• • •»

s

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'%o6 The twelfth Book of

* ■

PROP. X.

Every Cone is the third part of a cylinder having
the fame bafe with it JBCD, and the altitude equal.

If you deny it, then firft let fuch cylinder be
more than triple to the cone, and let the excefs
See the ft- be E. A prifm defcribed on a fquare in the circle
covd faun ABCD a is fubduple of a prifm defcribed upon
of tht * fquare about the circle, being equal to it and the
took cylinder in the heighth. Therefore a prifm upon
a fch 7 4. the fquare ABCD exceeds the half of the cyl. and
and 'cor c.likewife a prifm upon the bafe AFB, of equal
12 heighth to the cylinder, h is greater than the

b fch zj i halfof the fegmerit of the cylinder AFB, continue
and cor c! an equal bifeftionof the arches, andfubftraft the
* >. * prifms till the remaining fegments of the cylin-
der, namely at AF,FB,c3^* become lefs than the
folid E. Therefore the cyl. — fegm. AF,FB,&V.
c < a*. 1. (the prifm on the bale AFBGCHDI) c is greater
d fiyp ' ' than the cylinder — E (d the triple of the cone.)
tcor. 7 11. therefore the pyramide, e a third pait of the faid
'prifm (being placed on the fame bafe, and of the
fame heighth) is greatei than the cone of equal
heighth on the bale ABCD a circle i. e. the pare
gi eater than the -whole. Which is ahfurd.

But if the cone be affirmed to be greater than
thethird part ot the c linder, then let the excefs
be E. Detract the pyiamides from the cone, as you
did in the fiift part the prifms from the cylin-
der)

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EUCLlDE'i tkmenti. 507

<Jer, till fome fegments of the cone remain, con-
ceive at AF, FB, EG, &c- lefs than the iolid E.
therefore the cone — E (/ ; of the cylinder) -3 f hyf%
the pyr. AFBQCHDI (the cone - fegm. AF,FB,
gfr.) therefore the piiim triple to the pyramide
(viz. of equal heighth, and on the fame bale)
is greater than the cylinder, on the bafe ABCD,
the part than the whole, Wmch is nbfwd. W here-
fore it muft be granted, that the cylinder is equal
to triple of the cone,

PROP. XL.

Cylinders and Cones JBCBK, EFGHM> being
under the favie altitude, are to one another as their
lafes.JBCD, EFGH.

Let the circle ABCD. the cir. EFGH the cone
ABCDK.N. I fay N is equal to the cone EFG HM.

For if it be poflible,let N be "2 the cone EFG-
HM, and let the excefs be O. The preparation
and argumentation of the prec. Prop, being fup-
poled j then ihallO be greater than the fegments
of the cone EP,PF,FQ^y<:. and fo the folid N~3
the pyr. EPFQGRhSM. in the circle ABCD a aj o.%.ani
make a like poly gonous tigure nTBVCXDY. Be 1 poft.
caufe the pyr. ABVYK. the pyr. EFQSM b ;;b 6. 12.
ATBVY. the polyg.EPFQS*;:; the cir. ABCD.c cor.z.it;
the cir. EFGH d:: the cone ABCDK.N. e thence d hyp.
the pyr. EPFQGRHSM lhall be -3 N. contrary e 14.
to what was affirmed before. Again conceive N

V z tr the

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-08 The eleventh Book of

the cone EFGHM. and make the cone EFGH-
f hyp. andMO :: N. the cone ABCDK/:: the circle EF-
h miw-GH. ABCD. £ therefore O -3 the cone ABCDK5
Jioik which is ahfurd, as appears by what is Ihewn in

gi4.U the fii ft part.
* Therefore rather admit ABCD. EFGH :: tht
cone ABCDK, EFGHM. J!P7w& was to be dem.

The fame may be demonftrated t>f cylinders,
if cylinders and prifms be conceived in the place
of cones and pyramideF. therefore, &c.

Scbol.

Hence is gathered the dimenfion of all forts ofty-
Ihidtn and cones. The folidity of a right cylin-
a 1 Prop, der is produced of the circular bafe (a the di-
de dimenf. menfion whereof is to be learnt out of Archimc-
circ. des) multiplied into the heighth ; b whence
b 11. 12, in like manner that of every cylinder.

Therefore the folidity of a cone is produced
of the third part of the altitude multiplied
into the bafe.

PROP. XIL

A /?K

,Y P

Like cones and cylinders ABCDK, EFGHM, are
in triplicate proportion of that of the diameters TX9
PR, of their bafes ABCD, EFGH.
Let the cone A have to N triplicate proportion
of TXtoPR.I fay N isrrrthe cone^EFGHM. For
if it be poflible, let N be "3 EFGH^M. and let the
excels be O. therefore, N ~i thepyr.EPFQGR-
HSM. Let the axes of the cones be IK,LM, and
join the right lines VK, CK, VI, CI, and QM,

GMj

Digitized by Google

EUCLIDE'* Elementl 309

GM, QL, GL. Becaufe the cones are like, a a 24. def.
thence VI. IK QL. LM. but die angles VIK. ir.
QLM b are right angles, c therefore the trian-b 18. itfi
gles VIK,QLM are equiangular, d whence VC, 1 r.
VI QG. QL. alfo VI. VK QL. QM. there- c 6. 6.
fore of equality VC. VK QG.QM. e moreover d 4. 6.
VK.GK QM. MG. therefore again of equa-e 7, 5.
lity VC. CK QG. GM./ therefore the trian- f 5. 6.
gle VKC, QMG are like : and by the fame rca-
lon the other triangles of this pyramide are like
to the other of that $ g wherefore thepyramides g 9Jef.11
themfelves are like. /; But they are in triplicate hror.8.12.
proportion of that of VC to QG. k that is, of k 4. 6.
VI to RL, / or TX to PR. m therefore the pyr. 1 15. 5.
AIBVCXDYK. the pyr. EPFQGRHSM the mtyf. and
cone of ABCDK.N. n whence the pyr. EPFQ- fi.j.
GRHSM ""3 N. which is repugnant to whatn 14. j.
was affirmed before.

Again, take N c the cone EFGHM. make
the cone EFGHM. O :: N. the cone ABCDK
o .v the pyr. EPRM. ATCK p :: GQ. VC thrice o before,

q PR, TX thrice, but O r is -3 ABCDK. and in-
which was before fhewn to be repugnant, verfely*
Wherefore N = the cone EFGHM. Which? cor.S.n.
was to be deitu q 4. 6.

But forafmuch as what proportion foeverr 14. 5*
cones have, alfo cylinders being triple of them,
have the fame ; therefore cylinder to cylinder
fhallhave proportion triplicate of the diameters
Qf the bafes.

tJ J PROP.

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b in. 12.

A

E

IB

P

The twelfth Book of
PROP. XIII.

If a cylinder JBCDy be divi-
ded by a plane EF parallel to
the oppojite planes BCyAD,then
as one cylinder AF ED is to. the
other cylinder EBCF, fo is the
P axis GIto the axis HL

The axis being produced,
a take G1£-GI, and HL=
IHr=LM,and conceive planes

drawn at the points K, L,M,
parallel to the circles AD,'
BC, b therefore the cylinder
FD -= the cyl. AN. and the

c ir. 1*.

der EN is as multiple of the cylinder ED asthe
axis IK is of the axis IG. and in like manner
the cylinder FP is as multiple of the cylinder
EF, as the axis IM is of the axis IH. but as
t , IK is=,tV3lM, c fo is the cylinder EN=,c~,
d6.def. 5.-T3EP. d therefore the cylinder AEFD. the cyl.
EBCF::G1. IH. Winch wastobedem. ^

PROP. XIV.

Cones AEB, CFD, and cy-
linders JHyCK confifting upon
equal bafes JB,CD> are to one,
^another as their alt itudes ME,

. The cylinder AH, and the
l^axis EM being produced,
take ML-FN ; and at the
a t r. 12. point L draw a plane parallel to the bafeAB,*
bi3- 12- thenfhallthecyl.APUe-irCK.Abutthecyl.AH.

AP(CK) :: ME.ML (ft P.) Which was to be dem.
* apply 9. The fame may be affirmed of cones iubmple
mid 7« i2t of cylinders : * as alfg of prifms and Whites.

p r»

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EUCUDF; Ekmtnth

i11

prop. xv.

_ & — In equal cones BACy EDF,
K * Vh^f andcyhndeisBHyEK,thc bafes

cand altitudes are leavrocai
* BC. EF :: MD.LA.) And corns
and cinders, whofe bafes and
altitudes are reciprocal, are
JF equal one to another.

If the altitudes be equal
then the bafes are equal too, and the thing is ; evi-
dent. If unequal, then take away MO--LA.

I. Hyp. Then is MD. MO <g LA) 4^thc a 14.
cyl. EX: (e BH) EQi« the cir. BC. EF. Which b conjlr.

was to be dem. . „ ,_ W*

». fl>p. BC. EF e DM. OM (LA) /» the d i t. tU

cyl. l£ FQ* .v BC. EF b :: BH. tQ. Theretoree ifl.

the cylinder EK=BH. ^cb ™ lc° he dem' f " ' 1 z'
The fame argument may be ufed for cones, gi i . y.

Jin. i2«

PROP. XVI.

ko-S<

Xwo unequal circles
ABCGy D&F, having
the fame center M, to in-
fcribe in the greater circle
JBCG a folygonous figure
of equal, and even Jides%
which Jhall. not touch the
lejjer circle DEF.

Through the center
M draw the line AC cutting the circle DEF in
F, from whence raife a perpendicular FH. a di-» *•
vide the fcmicircle ABC into two equal parts ;
and the half thereof BC alfo ; and fo do tont>
nually. J till the arch IC become lets than the©
arch HC,from I let fall the perpendicular IL. It
is manifeft that the arch IC meaiures the whole
circle, and that the number of arches iseven,and

V 4 f u

Digitized by

5 I* T&e twelfth Book of

c fcb.16.4S0 that the fubtended line IC is the fide of the
polygon that may be infcribed without touch*
dror.T6.}.ing the lefler circle DEF. For HG d touches
e 28. r. the circle DEF. e to which IK is parallel, and
f 3 q.def.i. placed outwardly ; / wherefore IJC does not
touch the circle DEF j much lefs'do CI, CK,
and the other fides of the polygonott more te^
mote from the center. Which was to be done*

PROP. XVII.

\ •

EUCLIDE'* Element?. |i|

Let both the fpheres be cut by a plane paffing
by the center making the circles EFGH,ABCV$
and the diameters AC, BV drawn, cutting per-
pendicularly. In the circle ABCV, Hnfcribe the a 16. tzi
equilateral polygone VMLNC, &c. not touch-
ing the circle £ FGH : then draw the diameter
K*, and ereft DO perpendicular to the plane
ABC. by DO, and by the diameters AC, tye, cbn-*
ceive planes DOC, DON erefted, which (hall bs
i perpendicular to the circle ABCV, and fo in b ig, if;
thefuperficiesofthe fpheie make c the quadrants ccor.22^
DOC, DON. In which let the right lines CP, 1
FQ,QR> RO, NS, ST.T>, yO 4 be titted, equal, d 4. 1*
and of equal multitude with CN, NL, &i.
make the fame conftiu&ion in the other qua-
drants OL, OM, &c. and in the whole fphere.
Then I fay the thing required is done.

From the points P,S, to the Plane ABCV draw
the perpendiculars PX, SY, e which ihall fall on e j8. if.'
the feftions ACN*. Therefore becaufe both/the f IZ. te;
right angles PXC, SYN, ^and PCX, SNY infift- * u
ingonA equal circumferences, /are equal, the K 32.1.
triangles alfoPCX»SNYiareequiangular.Wherc- k conftr.
fore being PC k -SN, I alfo is PX^ SY, / and 1 26. u
XC^YN ; m whence DX^DY. n and therefore m ^ax.t;
DX. XC :: DY.YN. 0 therefore YX,NC are pa- n 7. 5.
xallels, but becaufe PX, SY are equal, and lincc 0 2. 6.
being perpendicular to the fame plane ABCV, p 6. 11.
they are alfo f parallels, q therefore YX, SP ihall q 33. 1.
be equal and parallel, r whence SP,NC, are pa- r p. lZm
rallel one to the other and fo the/ quadrilate- f 7. II#
Tal NCPS. and by the fame reafon SPQT,t 2,11.
TQRG, as alfo the t triangle y RO are fo ma-
liy planes. In like manner the whole fphere
may be fliewn full of fuch quadrilaterals and
triangles, wherefore the figure infcribed is a
polyedron.

From the center Du draw DZ perpendicular to u ir. 11.
the plane NCPS 5 and ioinZN,ZC,ZS,ZP. Be- % 4. 6.
caufe DN, NC x . : DY, YX« thence NC is/ cr YXy 14. $.

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rj l4 Tie twelfth Book of

(SP.) and likewife SP c TQ, and TQ.tr y R,
And becaufe the angles DZC.DZN^ZSjDZP
Z 3.ie/.n.r are right, and the hdesDC,DN,DS,DP,<t equal,
a i and DZ common, h thence ZC,ZN,ZS,ZP are e-
ii. qual one to the other; and confequently about
b 47. 1. the quadrilateral NCPSc a circle may be defcri-
c I5^e/.i.bed, in which (becaufe NS,NC,CP, are d equal,
d conftr. and NC tr SP) NC e f ubtends more than the qua-
e z8. 3. drant, /therefore the ang. NZC at the center is
f 33. 9. obtufe, g therefore NCq tr z ZCq (ZCq+ZNq.)
g 11. x. Let NI be drawn perpendicular to AC. there-,
h 32. 1. fore fince the angle ADN (b DNC-+DCN) k is,
k 9*ax. i. obtufe, the half of it DCN fliall be greater than
1 5. 1. > the half of a right angle j and fo that which re-
mains of the right ang. CNl fliall be lefs than it
n 19. i. 7i whence IN cr IC. therefore NCq (Nlq \ ICq)
o 47. 1. 0 ~ INq. therefore IN r ZC. and confequently
P47. 1. DZpcrDl.but thepoint I isq without the fphere
q cpr. 19. EFGH. and fo much more the point Z. wherefore
the plane NCPS, (whofe r next point to the cen-
t 47. 1. ter is Z) does not touch the fphere EFGH. And
if a perpendicular D / be drawn to the plane
SPQT, the point A and fo alfo the plane SPQT
is yet further removed fiom the center, which
is alfo true of the other planes of the polyedron.
Therefore the polyedron ORPQCN, &c. in-
scribed in the greater fphere, does not touch the
lefler. JH>ich was to be done.

. ..

V

I *

Corolt.

Hence it follows, That if in any other fphere a
folid polyedron, like to the a bovef aid f olid polyedron,
be in/cribed, the proportion of the polyedron in one
fibere to the polyedron in the other is ttiplicateoftfm
of the diameters of the Jpberes.

For if right lines be drawn from the centers of
the fpheres to ail the angles of the bafes of the
. . faid polyedrons, then the polyedrons will be di-
vided into pyr&nudes equal in number and like $.

whofe

Digitized by

EUCLIDE'j EltmmC $i£

whofe homologous fides are femidiameters of
the fpheres ; as appears, if the lefler of thfcfe
fpheres be conceived defcribed within the great-
er about the fame center. For the right lines
drawn from the center of the fphere to the an-
gles of the bafes will agree one td the other by
reafon of the likenefsof the bafes; and fo will
like pyramides be made. Wherefore fince every
pyr amide in one fphere to every pyr amide like
it in the other fphere, a has proportion triple toartf*&i£
that of the homologous fides, that is, of the
femidiameters of the fpheres ; and b as one py- b u*
ramide is to one pyramide, fo all the pyramides,
that is, the folid polyedrbn compofed of thefe,
are to all the pyramides, that is, the folid po-
lyedron compofed of the others ; therefore the
polyedron of one fpace (hall have to the poljrer
dron of the other fphere, proportion triple of
that of the femidiameters, c andfo of the dia-c 15.
meters of the fpheres.

Spheres BACyEDF^are in triplicate proportion one
to the other of that in which their diameters BC9
EF.

Let the fphere BAC be to the fphere G in tnpie
proportion of that of the diameter BC to the
diameter EF. I lay G - DEF. For if it be pof-
fible, let G be EDF. and conceive the fphere
G concentrical with EDF. In the Iphere of
EDF a inlcribe a polyedron not touching the a 17,
fphere G, and a like polyedron in the Inhere

BAC,

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3l"6 The "twelfth Bool of

bcor.ij. BAC Thefe polyedrons h are in triplicate pro>
12. portion of the diameters BC,EF.i that is,of the
c typ. jphere BAC f G. d Consequently the fphere
d 14, 5. G is greater than the polyedron infcribed in the

fphere EDF, the part than the whole.

Again, if it be poffible, let \he fphere G be

trEDF. and as the fphere EDF is to another
e lyf. in- fphere H, fo let G be to BAG, e that is, ia
uerf. proportion of the diameter EFtoBC. there-
to S* fore fince BAC / crH, we {hail incur the ab-

furdity of the firft part, wherefore rather the

fphere Q z=z EDF, Which was to be dent.

I

Corolla

Hence, As one fphere is to another fphere,
fo is a Polyedron defcribed in that, to a like;
folyedron defcribed in this.

The End pf the twelfth Boot;

: ad*

THE

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f -■

■

t HE THIRTEENTH BOOK

OF

EUCfclDE'* ELEMENTS.

■ -•

-

PROP. I.

w

F a right line z be divided according to ex-
treme and mean Proportion (z. a :: a. e.) the
fquare of the half of the whole line z, and
of the greater ferment a, as one line is quin-

tuple to that which is defcribed of half of

that whole line z.

I lay Q. a 4 i z
xjQ;}z,<i thata 4. il
A E is, aa i zz zab 3. ax.xi
~z zz. -h i zz. * or aa -\- za = zz. For ze -| c 2. 2.
za <: = zz. and ze J = aa. e therefore aa -t za d hjp. and
-= zz. Which was to be denu 16-6.

PROP. II. *****

Set the 1. SWetf*.

If a ^ne r z a itf i« jfwer quintuple to a
fegment of it f*lf ± z. the line double of the faid
fegment (z) being divided according to extreme and
meanproportion, the greater fegment is (/) the other
part of the tight line at firft given {■ z + a.

I fay z. a :: a. e. For becauie by the hyp. * aa * 4. 2.
-4. i zz za — zz - 1 zz ; oraa-*- za = zza az.2.
= ze + za, b thence fhali aa be = ze. c where- b 3. ax. I.
fore z. a.va.e. Which was to be denu t c 17.6.

\

r

PROP.

y

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73* thirteenth Book of

PROP. III.

If ng/;* /iwe z ie divided according to extreme
and mean proportion (z. a :: a.e.) the Line made djf
*fa /e/j fegment e half of the greater fegment a,
i* in ^0jw quintuple to the fquare9 which is de-
ferred of the half line of the greater fegment a.

^ — Hay Qre-Hi-ar:?

Q: {• a. a that is ee

v

& 44 2.
b$. AT*

C 2.

d iyp. tfwiee+ea £ = ze 4 -=aa. WJricb was to be dem.
17.6.

PROP. IV.

i aa-H eaz=aa h--^ aa.
h or ee -h ea=aa. For

a 4. I,

b 3. z.
c 17. 6.
dz.ix.

, If a right linez he cut according to extreme and
mean proportion (z. a :: i. e.) the fquare made of
the whole line z, and that made of the leffer fegment
e, loth together jrc triple of the fquare made of
the greater fegment a,

I fay zfc -J- ee =r 5
aa. a or aa -t- ee z

.. f ae-f fe^j aa. For

A. E ae -h ee i = ze c =

aa. i therefore aa -1- z ae ee = 5 aa. JPifci'
was to be dem.

D

A

1-

C
1

PROP. V.

B

If a right fine AB be cut
according to extreme and
mean proportion in C9 and
a line JD> equal to the greater fegment BC9 added
to it, the whole right line DB is divided according to
extreme and mean proportion $ and the greater feg-
ment the right line AB given at the beginning.

For becaufe AB. AD a ;: AC.CB. and by inver-
sion A I?, AB::CB. AC. therefore by compofition
t)B. AB :: AB.AC (AD.) Which was to be dem.

■ Schoh

Digitized by Google

EUGIIDE'j Ekttiini?. jij

ScboL

But if BD. BA :: BA. AD. then fhall be BAi
AD:: AD. BA — AD. For by divifion is BD—
BA (AD.) BA :: BA - AD, AD. therefore in*
verfeJy BA. AD :: AD. BA - AD.

PROP. VL

B If a rational right I'm*
AB be cut according to ex-
treme and mean proportion
in C, either of the fegments (ACfiB) is an irrational
line of that kind which is called avotome or refidual.

.To the greater fegment AC, a add AD^i- AB.a 3. u
]l therefore DCq = $ DAq. c therefore DCqxu b 1. t}«
DAq. confequently d fince AB, e and fo the c 6. 10.
half thereof DA are £, likewife DC is p. But d hyt.
becaufe 5. 1 :: not Q. Q. / thence is DC Tl DA. e fck iz.
g therefore DC — AD, that is, AC, is a refi- 1©#
dual line. Further, becaufe ACq h — AB xf 9. 10.
BC, and AB is p. k likewife EC is a refidual g 74* 10;
Which was to he dem* h 17. 6.

PROP. VII.

if three angles of an equilateral Pentagone ABCDE
whether they Mow in order, (EAB, ABC, BCD,)
or not, (EAB, BCD, CDE) he equql, the Pentagon*
ABCDE Jhall he equiangular*

Digitized by Google

% 2$ , The tbirtemth Book of

Let the right lines BE, AC, BD, be fubtend-
cd to the equal angles in order.
Being the fides EA, AB, BC, CD, arid the in-
a hyP* eluded angles a are equal, therefore lhall the ba-
b 4. r. Ces BE, AC, BD, c and the angles AEB, ABE,
c 4. and $. BAC, BCA, be equal, d wherefore BF = FA. c
1. and confequently FC—FE 1 therefore the trian-

d 6. 1. , gles FCD,FED, are equilateral one to the other i
ei.ax. i./ whence the angle FCDrrFED. g conlequently
f 8. 1. the angle AED^=BCD. In like manner the ang.
g 2. ax. 1. CDE is equal to the reft ; wherefore the pen-
tagone is equiangular. Winch was to It dem.
But if the angles EAB, BCD, CDE, which
li 4. r. are not in order, be fuppofed equal, b then fhall
k 5. 1. the angle AEB be = BDC, and BE = BD. k
1 2. ax, and thence die ingle BED = BDE. / confe-
quently the whole angle AED «= CDE. there-
fore becaufe the angles A, E, D, in order, are
equal, as before, the pentagone lhall be equian-
gular. Winch was to be dem.

PROP. VIII.

If in an equilateral and e~
quiangular Pent agbneAB CDE,
two right lines BD, CE, fub-
tend two angles BCD> CDE,
following in order, thofe lines
do cut one another according to
extreme and mean proportion,
and their greater fegments BF
or EE are equal to the fide of the pentagone BC.
a 1 4. 4. a Defcribe about the pentagone the circle ABD.
b 28. 1. * The arch ED is = BC, c therefore the angle
C 27. \. FCD rr FDC, d therefore the ang. BFCrr 2 FCD
d 32. 1. (FCD + FDC. But the arch BAE b is =2 ED,
c 22. 6. and confequently the angle BCF e = 2 FCD=
f 6. 1 . BFC. /wherefore BF = BC. Which was to be dem '.
27. Moreover becaufe the triangles BCD, FCD, are
4. 6. g equiangular, h therefore BD.DC (BF.) :: CD.

(BFJf

i

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EUCLIDE'* Elements . %n

(BF.) FD. and likewife EC.EF :: EF. FC. Winch
wat to be dem.

PROP. IX.

if the fide of an Hexagone
BE, and the fide of a Deca-
goneAB, both defcribed in the
fame circle ABC, be added ro-
gether, the whole right line
AE is cut according to ex-
treme and mean proportion
(AE. BE :: BE. AB.) and
the greater fegment thereof is the fide of the Hexa-
gone BE.

Draw the diameter ADC, and join the right
lines DB, DE. Becaufe the angle BDC a zz. az /;y* and
BDA. and the angle BDC b — z DBA (DAC+17 ,
DBA) thence mail DBA (b DBE BED) e beb \z\ t.
r= z BDA d=iz BDE. whence the angle DBA c 7. ax* 1.
brDABe^ADE. Therefore the triangles ADE,d 5. 1.
ADB, are equiangular : / wherefore AE. AD (ge wax. r.
BE):: AD. (BE.) AB. Which was to be dem. f4.6.

CoroU. gf9MJ4'

Hence*, If the fide of an Hexagone in a circle
be cut according to extreme and mean propor-
tion ; the greater fegment thereof lhall be the
fide of the Decagone in the fame circle.

PROP.

The thirteenth Book if

PROP. X.

EES A

7. ax.
b hyp. and
J. ax.
c JJ 6..
d 2.0. g«

f JI. i.

g 4« 6*

k 17-
m 4« !•
n 2,7. 3.

p 4. 6.
q 17. 6.
r 2. 2.

f 2. a*.

Jf an equilafe*
ral Pentagone J-
BCDE be dejcri-
led in a circle
J£CE9 the fide oj
the Pentagone A
contains in tow
"both the fide of a
hexagone FBy and
the fide of a deca-
goneJH defcribeA
m the fame circle.
Draw the dia-
meter AG. and bifeft equally the arch AH in
K. and draw FK, FH, FB, BH, HM.

The femicircle AG — the arch AC fl = AG

- AD. that is, the arch CG = GD b = AH

— HB. therefore the arch BCG = 2 BHK 5 c
and fo the angle BFG — 2 BFK. i but the an-
cle BFG = 2 BAG. e therefore the angle BFK
5= BAG. Wherefore the triangles BFM, FAB, f
are equiangular, g whence AB.BF :: BF. KM. h
therefore ABx BMa^BFq. Moreover, the an-
gle AFK fe = HFK, and FA = FH. m where-
Fore AL= LH. m and the angles FLA, FLH
are equal, and fo right angles, therefore the an-
gle LHM m — LAM n = HBA. therefore the
triangles AHB, AMH, 0 are equiangular, where-
fore AB. AH :: AH. AM. a therefore ABx AM
== AHq. So that feeing ABq r = AB x BMh-
AB x AM,/ thenceABq = BFq-+AHq. Wlmh
was to be dem.

t^oll.

1. Hence, a right line (FK) which being drawn
from the center (F) divides an arch (HA) into
two equal fegments, does alfo divide the right

line

*

EUCLIDE'j Element?.

line (HA) fubtending that arch, perpendicularly
into two equal fegments.

2» The diameter of a circle (AG ) drawn from
any angle (A) of a pentagone, does divide equal-
ly in two both the arch (CD) which the fide
of the pentagone oppofite to that angle fub-
tends, and alto the oppofite fide it felf (CD)
and that perpendicularly.

Schol.

Here, according to our Promife, we Jball lay down
a ready praxis oft he n. Prof, of the 4. Book,

• At* > % •

Problem.

r

To find out the fide of a pentagone to he inferiied
in a circle JDB.

Draw the diameter AB, to which ereft a
perpendicular CD at the center C divide CB
equally in E. and make EF = ED. then DF
fhall be the fide of the pentagone. a 6. z.

For BF x FC ECq ar^ EFq h — EDq c = bconftr.
DCq 4- ECq. d therefore BF x FC -|- DCq or c 47. 1.
BCq. e wherefore BF. BC :: BC. FC. therefore d ax.
fince BC is the iide of an hexagone, / FC fhall e 7- •
be the fide of a decagone. Confequently DF f 9. 1$.
/; — V DCq \ FCq g is the fide of a pentagone. g 1 o. 1 \
VHnch was to he done, h 47. f •

X z PROP.

324

4

The thirteenth Bool of

? ro p. 11

* i«. 6.

a ror. ic.

b 32. i.
c 4. 6.
d 1$.
e 18. 5.
t* 21. 6.

g i-H*

h 9. 10.
k74. 10.
1 9. 10.
m cor. 8.£.
fl?2/i 17.6.
n 95* io«

If in a circle ABCD,
wbofe diamr+er is ratio*
nal AG, an equilateral
fentagone he vifcribed
ABCDE \ the fide of the
fentagone AB is an ir-
rational line of that kind
which is called a minor
line.

Draw the diameter
BFH, and the right lines AC, AH ; and * make
FL = i of the radius FH ; and CM = i CA.

Becaufe the angles AKF, AlC, are a right an-
gles, and CAI common, the triangles AKF,
AlC, are h equiangular : c therefore CI. FK c ::
CA. FA (FB) d ;; CM. FL. therefore by permu-
tation FK. FL :: CI. CM d :: CD. CK (1 CM )
and fo by e compolition CD -+ CK. CK .*.' KL.
FL. /confequently Q: CD -4. CK (g $ CKq.)
CKq :; KLq. FLq. therefore KLq = 5 FLq.
wherefore if BH (p) be taken 8, FH (hall be 4,
FL 1, and FLq 1, BL 5, and BLq if, KM 5,
by which it appears that EL and KL are p h
k and fo BK is a refidual, anH KL its congruent
or adjoining line, but being BLq — KLq = 20,
/ thence BL -a. V BLq - KLq. m whence BK
(hall be a fourth refidual line. Therefore becaufe
ABq m is — . HB x BK, n lhall AB be a minor
line. Which wnu to be dew.

■

PROP.

^1

I

Digi^zed by Googl

EUCLIDE'* Elements.

prop. am.

32?

J/ in a cwfe ^5£C d»
equilateral triangle ABC he
inftribed, the fide of that tru
angle AB is in power triple to
the line AD drawn from D
IC the center of the circle to the
/ circumference.

The diameter being ex-
,JS< tended to E, draw BE. Bc-

caufe the arcfc BE a = EC, the arch BE is the a cor. 10.
fixth part of the circumference, b therefore BE 15.
= DE. tonce AEq c — 4 DEq ( 4 BEq) d = b for.15.4.
ABq i~ F^cl (r* ADq.) e confequently ABq = c 4. z.
3 ADq. Which was to be dem. d 47. i»

CorolL e 3. ax. r.

1. AEq. ABq :: 4. j.

ABq. AFq :: 4. 3. / For ABq. AFq :: AEq.f cor. 8.6,
ABq. **• 6.

j. DF — FE. For the triangle EBD ^ is e- gcor.i^.
quilateral, £ and BF perpendicular to ED. bbcor.W.
therefore EF = FD. . 1

4. Hence, Af* = DE h- DF = 3 DF.

»-.••-. . *

PROP, XIII,

To defcrile a pyramide Ecfcl, and comprehend it
■ ill 4 $/!ere £?'t;e?i : rfwd to demonftrate that the dui-
viettr of the fjfhtrt AB is in power fefquialter of the
' fide EF of the pyramide EG FT.

X 3 About

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Tie thirteenth Book of

9 io. 6. About AB defcribe thefemicircle ADBj a and
let AC be = z CB. from the point C ereft the
perpendicular line CD; and join AD, DB, then
at the interval of the radius HE = CD defcribe
bcor.i $.4, the circle HEFG, wherein inferibe the equilate-
c 12. 11, ral triangle EFG from He ereft IH=CA perpen-
d 3. 1. dicular to the plane EFG. produce IH to K, d to
that IK - AB \ and join the right lines IE,IF,IG,
Then EFGI fliall be the pyramide required.
For becaufe the angles ACD,IHE,IHF,IHGF,
e conllr. e are right angles ; and CD,HE,HF,HG e equal,
f 41. r. e and IH AC y /therefore AD, IE, IF, 1(3
g 20. 6. be equal among themfclves. But being AC

h z. ax. (* CB) CB g :: ACq. CDq. then flia!l ACq be =
k 12, i?# 2 CDq. therefore ADq / = ACq + CDq* =5
1 1. axli. J CDq = 3 HEq k = EFq. / therefor? AD,EF,
IE, IF, IG are equal, and lo the pyramide EF-
GI is equilateral. But if the point C be placed
upon H, and AC upon HI, the right lines AB,

m 8. ax. IH» m agreet 215 being equal. Wherefore
the femicircle ADB being drawn about the axis
ni5.A/.r,AB or IK n fhall pafsby the point E,F, G, *
* j 1. def.*n& fo the pyramide EFGI fliall be iftferibed in
ir. a fphere. winch was to le done.
o cor. a 6. Alfo it is manifeft that BAa. ADq 0 ;: BA.
p confir. AC $ :: J. 2. Which was to he 4cm.

CoroU.

f. ABq.HEq:: a 2. For if ABq be put 9,
then ACq (EFq) mall be 9, confequently HEq
q 12. 1 j. fliall be 2*

2. If L be the center, then fliall AB.LC z6.u
For if AB be put f, then AL (hall be 3. r and
thence AC a. wherefore LC fliall be 1. Hence
1 conjlr. 3. AB. HI :: tf. 4 :: 3. 2. whence
4. ABq. Hlq :: 9* *

prop:

Digitized by

EUCLIDE'* Elements.

PROP- XIV.

Djpherc is in power double
of AC the fide of that
Ottacdron.

To def exile an 03ae-
Jron KEFGDL, end
* comprehend it in the
given fyhere^ wherein a
Pyr amide is : and to
demovjlrate that AH
the diameter of the

About H defcribe the femicircle ACH. and
from the center B ered the perpendicular BC. •
draw AC, HC, then upon ED:=rAC. a make the a 46. 1
fquare EFGD, whofe diameters DF, EG, cut in
the center I. from I draw IL=AB£perpendicu- b 12. 1
lar to the plane EFGD. produce IL,c till IK=IL. c J. U
and join KE,KF,KG,KD,LE,LF,LG,LD ; then
fhall KEFQDL be the Octaedron required.

For AB,BH,FI,IE, &c. being feinidiameters of
equal fquares are equal one to the other, d whence d 4. 1.
the bafes LF, LE, FE, §rV. of the right-angled
triangles LIE, LIF, FIE, &c. are equal, and
confequently the eight triangles LFE,LEG,LGD,
LDE,KEF,KFG,KGD,KDE, are equilateral, ee ij.dt
and make an Octaedron, which may be inferi- II.
bed in a fphere, whofe center is I, and IL or
AB the radius, (becaufe AB, IL, IF, IK, grV. fi conjlu
are equal.) Which was to be done. Moreover, it
is evident that AHq (LKq) g = z ACq (2 g 47. 1.
LDq.) Winch was to be dem.

1. Hence it is manifeft, that in theo£taedron .
the three diameters, EG, FD, LK do cut one the
other perpendicularly in the center of the fphere.

2,.1 Alfo that the three planes EFGD, LEKG,
LFKD are fquares, cutting one another perpen-
dicularly.

Cor oil.

x4

3. The

328 The thirteenth Book of

3. The Oclaedron is divided into two like
and equal pyramides EFGDL, and EFGDK,
whofe common bafe is the fquare EFGD.
f J. ir. 4. Laftly, It follows that the oppolite bafes of
the pftaedron are parallel one to the other.

PROP. XV.

(lvf ^_ Jj To defcrihe a cult

kt ^1 EFGHIKLM^ and

comprehend it in the
fame fphere wherein
the former figures
were 3 and to demon-
ftrate that AB the

F diameter of the

fphere is in power triple to EF the fide of that cule.
a 10. 6. Upon AB defcribe a femicircle ACB \ a and
make AB- 3 DA, from Draife the perpendicular
DC, and join BC and AC. Then upon EF^AC
J> 46. ir. I make the fquare EFGH, upon whofe plane let
the right lines EI, FK, HM, GL, ftand perpendi-
cular, being equal to EF, and conneA them
with the right lines IK, KL, LM, IM; The
folid EFGHIKLM is a cube, as is fufficiently
apparent from the confirmation.

In the oppofite fquares EFKI, HGLM, draw
the diameters EK,FI,HL,MG, by which let the
planes EKLH,FI,MG be drawn,cutting oneano-
c cor. jo, ther in the line NO. which c fhall divide equally
t 1. in two parts the diameters of the cube EL,FM,GI,
d 1 5. def HK, in P the center of the cube, d therefore P
x%, and 1 4, fhall be the center of a fphere pafling by the an-
def 11. 'gular points of the cube. Moreover,kLq e^EKq
C47. 1. -KLq,*T= JKLq,/or?ACq1butAB?.A6q^::
f conjlr. BA.DA f:: /; therefore A B - EL. wherefore we
g cor. 8. 6, havc ™ade a cube, &c. Which was to be done.
h 14. ^. Cor oil.

I. Hence it is rrpn ifeft, that all the diameters
of the cube are equal one to another, and do
equally bi feci: one another in the center of the
fphere. And by the fame means the right lines

whiclj

Digitized by Google

EUCLIDE'* Element?. fig

which conjoin the centers of theoppofite fquares
are equally bife&ed in the fame center.

z. The diameter of a fphere comainsin power
the fide of a tetraedron aid of a cube, viz, ABq
k-=z I BCq •+ m ACq. k 47- *•

PROP. XVI. 1

m

To Aftrsle an Ico/aedron
ZGH1KFYVXRST , ™-
r omftf/i tl iw the fthere, wherein
were contained the /ore/aid /o-
ftfe : A*i to demonftrate thatC
FG the fide efthe Ico/aedron u
that mathonal line which is
called a minor lines -

Upon ABth* diameter of a
fphere defcribe the lemiciicle
ADB; and • make ABtfsBC.
then from C erett CD perpen-
dicular, and draw AD and
BD. At the diftance EF ^
BD defcribe the circle EFK-

KG 5 A

-

Digitized by Google

3$o 72tf thirteenth Book of

b llt-4# NQ;iwherein infaibe the equilateral pcntagofte
FKIHG. Divide equally in two parts the arches
FG,GH,&V. and join the right lines FL,LG,g? c.
c 12 ii being the fides of a decagone. Then r ereft EQ,
LR,MS,NT,OV,PX equal to EF, and perpendi-
cular to the plane FKNG ; and conneft RS,ST,
TV, VX, XR ; as alfo FX, FR, GR, GS, HS,
ST, HT, IT, ir, KV, KX. Laftly, produce FQ,
and take QY = FL, and EZ = FL. and con-
ceive the right lines ZG, ZH, ZI, ZX, ZF to
be drawn ; as alfo YV, YX, YR, YS, YT. Then
I fay the Icofaedron required is made,
d eonftr. For becaufe EQ,LR,MS,NT,OV,PX, are d e-
e 6* ii* equal and e parallel,alfo t hofe lines that join them
f 22. i. EL,QR,EM,QS,EN,QT,EO,Qy, EP,QX, /are
equal and parallel. And thence likewifeLM (of
FG) RS,MN,ST,fiiV. areemialoneto the other,
s 1 5. ii. g therefore the plane drawn i>y EL, EM, &c. is
equidiftant from the plane palling by^R, QS ,
h uitf.Z.&c. h and the circle QXRSTV drawn from the
center QJs equal to the circle EPLMNO \ and,
RSTVX is an equilateral pent^gone. But EF,EG,
TZHi&c and ■QX,QR,QS,&c. being conceived to
Jk 47. t. be drawn} then becaufe FRqJfc^FEq+LRq, / or
1 eonftr. EFq m = FGq, n therefore FR, FG, and fo all
Ui io. i2..RS,FG,FR,RG,GS,GH,&V.lhall be equal one
n/dU&i.to the other, and confequently the ten triangles
end i. ax. RFX,RFG,RGS, &c. are equilateral and equal,
o cot id* Moreover, becaufe XQY isaorightangle,there-
11. fore XYq p = QXq + QYq q = VXq or FGq,
p 47. i. wherefore XY,VX. and likewise YV,YT,YS,YR,
q io. i j. ZG, ZH, &c. are equal. Therefore other ten
triangles are made equilateral and equal both

• an Icofaedron is made.

Moreover, divide equally EQ^in a, draw the
f is4£f .fight lines *F,*X,*V; and becaufe QX r=Qyf
and *Q. the common fide, and FQX, EQV are
f 4» i% light angles, / therefore fliall *X be = ctV ; and
by the fame reafon all the lines aX, «R, ctS, *T,

Digitized by G

EUCLIDFx Events. ||i

tV, aF, ctG, *H, «I, aK are equal. But becaufet % r j#

fore the fphere, whofe center is*, and *F thezij. y#
ray, fliall pafs by the i z angular points of the
Icofaedron,

Laftly, Becaufe Z*. *E z ZY. QE ; a and foa zz. &
Z*q, *Eq ZYq. QEq, i therefore ZYq = 5b 14. 5.
QEq, or $ BDq = : but ABq. BDq c :: AB.BC :: $. c cor. 8.6*
1. d therefore ZYsr AB. Which was to he done^ d r. ax< u

Therefore if AB be put p, rthen EF =V ABq e/i& iz.
fliall be alfo p. and confequently FG the fide of ie.
the pentagone, and likewife of the Icofaedron, /f 11. ij.
is a minor line, Whuh was to U dem<

CoroB.

U From hence is inferred, that the diameter
of the fphere is in power quintuple of the
feraidiameter of the circle encompaffing the five '
fides of the Icofaedron*

z. AUb it is manifeft that the diameter of
tht fphere is compofed of the fide of a hexagone,
that i»> of the femidiameter, and two fides of
the decagpne of a circle eacpmpaffing the five
fides of tltut Icofaedron.

3. It appeats likewife that the oppofite (ides
of an Icofaedron, fuch as RX, HI, are parallels.

for RX a is paralk to LP. * paiall. to HI* j j J. 1.

PROP.

s

Digitized by

The thirteenth Book of
PROP. XVII.

1 .

-v.

1 T

I

»

I

l<3

» t t »

To defer lie a Dodecaedron, and comprehend if in
the fphere wherein the former figures were comprehend*
ed : and to demon/Irate that the fide RS of the Do~
iecaedron is an irrational line oftttatfl* nfiic/j is
called an Jpotome or refidual line.

Let AB be a cube inferibedin the given fphere,
and let all the fides thereof brdivided eaually in
the points E,H,F,G,K,L,6>V. and join the right
lines KL,MH,HG,EF. *make HI.IQ::IQ.QH;
and take NO, NP, —IQ.. then ereft OR* PS, per-
pendicular to the plane DB, and QT to the plane
AC, and let OR, PS, QT, be equal to IQ, NO,
NP, whence DR,RS,SC,CT,DT, being conneft-
cd, DRSCT /ball be a pentagone of the dode-
caedron required. For drawNv parallel to OR,
and having drawn NVout as far as the center of
the cube X, join the right lines DS,DO,DP,CR,

CP,

Digitized by

CP,HV,HT,RX. Becaufe DOq *-=DKq (JtfNq) a 47. 1.

-+KOq c r=L 1 ONq (2 ORq) d thence DRq ±=b 7. ax. f.
ORq e s= OPq, or RSq. therefore DR- RS. t 4. 13.
/ the fame realbn DR,RS,SC,CT, TP are equal, d 47. x.
ut becaufe OR / is = and g parallel to PS, e 4. t.
therefore RS,OP, and confequently RS, DC fhall 1 confir.^
be aifo parallels. A therefore thefe with them 11.
that conjoin them DK,CS,VH,are in one and the g $ 3. r.
lame plane. Moreover, becaufe Kfl. IQ h :: IQ.fi 9. 1.
(TQ.) QH k :: HN. N V. and both TQ,HN, and k 7. iU
QH,NV & are perpendicular to the fame plane, 11 6. ir,
and fo likewife parallels, 01 THV fhall be a right m 32. 6.
line, ft therefore the Trapezium DRSC, and then i>andz*
triangle DTS are in one plane extended by their,
right lines DC,TV . 0 therefore DCTSR is a pen- o J. 1?.
tagone, and that alfo equilateral, by what is
fhewn already* Furthermore, becaufe PK. KN ;:
KN.NP ; and DSq; = DPq-4-PSq (PNq) = p p 47. t.
DKq-4-PKq-^NPq^ thence DSq—DKq-H^KNqq t.ax. u
=4DKq (4DH0J r-DCa. therefore DS = DC. and 4. 1 J.
whence the triangles DRS, DCT, are equilateral r 4. 2.
one to another,/therefore the angle DRSr=DTC,f 8. I.
therefore the pentagone DTCSR is alfo equian-
gular. Moreover, bScaufe AX, DX, CX, &c. are
leinidiameters of the cube, t thence is XN^IH t 1$. 1 j.
or KN, n and fo XV — z KP; wherefore becaufe u 1. ax. I«
RVX,is a or right angle, z thence RXq — XVq % 29. r.
fcVq (NPq) :riKPq+NPq* = gKNq£ rrAXqz 47. 1,
or t)Xq, &c. therefore RX, AX,DX, and by the a 4.13.
lame reafon XS,XT,AX,are equal one to another, b 1$. 1 ?•
i^nd if by the fame method, whereby the penta-
g6ne DTCSR was made, twelve like penta-
gones, touching the twelve fides of the cube,be
made, they iliall compofe a Dodecaedron; and a
fphere paffing by their angular points, whofe
radius is AX or RX, fhall comprehend that Do-
decaedron. Which was to be done.

Laftly, becaufe KN.NOc:: NO. OK, dthtnctc conftr.
KL.OP ::OP.OK+PL, Therefore if AB th^dia- d 15. J.
meter of the fphere be fuppofed then fhall

Digitized by Google

The thirteenth Book of

AB

Z IV 1?# KL $ S5 a/— /be alfo p. £ whence OP or

1 RS the fide of the dodecaedron (hall be a refi-

g 6. i?. dual line* JPfoVJ was to he dm*

CoroU.

From this dcmonftration it follows, I. that if
the fide of a cube be cut in extreme and mean
proportion, the greater fegment fliall be the fide
of the dodecaedron infcribed in the fame fphere.

2. If the leffer fegment of a right line, cut in
extreme and mean proportion, be the fide of the
dodecaedron, the greater fegment lhall be the
fide of the cube infcribed in the fame fphere.

It is manifeft alfo, that the fide of the cube
is equal to the right line which fubtends the an-
gle of a pentagone of the dodecaedron, infcri-
bed in the fame fphere.

a to. r .

b to.6.

PROP.

c jo. 6.

Q To find out the
^ fides of the pendent
five figures j and com*
fare them together.

Let AB De the
diameter of the
fphere given, and
A EB the femicircle,
and let AC a at i
AB, andAD*=f
AB then ereft the
perpendiculars CE,
DF,^ndBG— AB.
join AF, AE, BE, BF, CG 5 an4 let fall the
perpendicular HI from H $ and CK being taken

Sual to CI, from K ereft the perpendicular
t, and join AL. Laftly, € make AF. AO ::
AO. OF.

There*

Digitized by Gc

EUCLIDFi Element^

Therefore j. z d :: AB. BD e :: ABq. BFq thed tonftr.
fide of a Tetraedron and 2. i ::a AB.AC :: ABq.e cor. 8.6.
BEq. /the fide of an O&aedron. f 14. ij.

Alio 3. 1 d :: AB. AD e :: ABq. AFq. g theg 1$. 13.
fide of an Hexaedroru h conjlr.

Moreover, becaufe AF. AO h :: AO. OF. fck cor. ij+
thence ihall AO be the fide of a Dodecaedron. 1$.
Laftly, BG, (2, BC.) BC / :: HI. IC. m therefore 1 4. 6.
HI = z CI n z=z KI. therefore Hlq 0 — *Clq. m 14. ^
q confequently CHq p = 5 CIq. t therefore n t o»/!r.
ABq=: S Klq. r therefore KI or HI is a ra-o 4.2,
dius of a circle enclofing the pentagone of an p 47. r%
Icofaedron ; and AK or IB r is the fide of a de- q 1 5. j#
cagone infcribed in the fame circle, / whence r cor. 16V
AL ihall be the fide of a pentagone, * and alfo 1 3.
the fide of an Icofaedron. Whereby it appears f I0. ij*
that BF, BE, AE are J -g -. and AL, AO p nx,t 16. 15.
and BF cr BE,and BE,AF,and AF cr AO. And u 1. ax. u
becaufe 5 AFq = ABq u = $ KLq, and AF x x 4. ax. u
AO ir AF x OF, x and fo AF x AO AF x y i.x.
OF cr 2 AF x OF, r that is, AFq cr z z AOq. z 17. d.

thence ihall $ AFq (5 KLq) be cr 6 AOq, a 47*1,
confequently KL cr AO, and much rather AL
cr AO.

That we may exprefs the fides in numbers ; If
AB be fuppofed J 60, then, reducing what is
already fliewn to fupputation, BF =r *J 40, and
BE = V 50, and AF V *o. Alfo AL = ?o

— V i8o(forAK=r^iJ - V J.a*iKL(HI)

= V u;j Laftly AO := jo — V Soo (v *S

- 1/ J.)

■ •

ScioL

gj6 the thirteenth Book\of

r • " *

' » • •.. ♦ . . j

// is very Apparent that lefides the five afore f aid
figures, there cannot be defcribed any other regular
folid figure (viz. fnch as may le contained under oy
dinate and equal plane figures.)
For three plane angles at leaft are required to
r the conftituting of a folid angle ; a all which
S?#r iL; muft be left than four right angles, h but 6 an-
to bee J cm. leg of an equilateral triangle, 4 of a ftyiare, and '
3** *• 6 of a hexagon, do feverally equal 4 right in-
gles ; and 4 of a pentagon, 3 of a heptagon, 3 of
an oftagon, &c, do exceed 4 right angles
Therefore only of J, 4, or 5 equilateral triangles,
of 3 fquares cr 1 pentagones, it is poffible to
make a folid angle* Wherefore befides the five
above-mentioned, there cannot be any other
regular bodies. V " • ' -

' ./Oat of P. Hcrigon.

1 '

the Proportions of the Jphere and the five regular
figures inferibed in the fame.

Let the diameter of the fphere be 2. then fhall

The Periphery or circumference of the great-
er circle, be tf, 18318*

The fuperficics of the greater circle, J,

The fuperficies of the fphere, iz, 5#37*

The folidity Of the fphere, 4, 1879.

The fide of the Tetraedron, 1,6x299.

The

* Digitized by G<

EUCLIDE*/ Elements:
The fuperficies of the tetraedron, 4, 6188.

The folidity of the tetraedron, a, 1 5 1 3 z.

The fide of the hexaedron, 1, 1547.

The fuperficies of the Hexaedron, 8.
The folidity of the hexaedron, 1, 5396,

The fide of the O&aedron, 1, 4^21.
The fuperficies of the oftaedron, 6, 9282.
The folidity of the o£taedron, 1, 33333.

The fide of the Dodecaedron, o, 71364,,
The fuperficies of the dodecaedron, 10, 5^62*
The folidity of the dodecaedron, «2, 78516.

The fide of the IcoEaedron, 1,0514$.

■

The fuperficies of the Icofaedron, 9, 57454,
The folidity of the I cofaedron, 2, 53615*

?3*

The -thirteenth Book ofy &C«

If jive equilateral and equiangular figures, like
thtfe in the J chevies heneath, he made of Pafer, and
tightly folded, they will repefent the five regular

todies.

1

6

The End of the thittttnth Booh

THE

Google

"A.

THE FOURTEENTH BOOK

OF

EUCLIDE', ELEMENTS*

PROP. L

Perpendicular line BP
drawn from B the cen-
ter of a circle ABC
to JSC the fide of a
fentagone infcrihed 4
in the J aid circle, is the half of
thefe two lines taken together, viz.
of the fide of the hexagone BE,
and the fide of the decagone EC
infcrihed in the fame circle ABC.

Take FG = FE, and draw CG : a Then CE a 4. i;
is — CG. therefore the angle CGE h = CEG b 5. 1.
I = ECD. therefore the angle ECG c = EDC c 31. 1.
d — i ADC e~{- CED (; ECD) /confequent- d hyp. a\
ly the angle GCD = ECG^EDC. £ wherefore 33. <5.
DG-GC(CE0 therefore DFnCE(DG)-fEF=e 20. J.

DE h- CE,

Winch was to he dem.

> 1

f 7. ax.
g&x.

PROP. II.

D

C If two right lines AB,BEf
— he cut according to extreme
F and mean proportion (AB.

AG :: AG. GB. and BE.

BH :: BH. HE.) they fi all
be cut after the fame manner, viz. into the fame
proportions {AG. GB :: BH. HE)

G
-1-

H
-1.

B
•1-
E
-I-

Y z

Take

7ao The fourteenth Book of

a 17. 6. Take EC— BG 5 and EF=EH. Then ABxEG
b 8.1. is.^^"$ AGq. wherefore ACqfc—4 ABG+AGq
c z. jr. i.ciBGq. In like manner fhall DFq be =-= 5 DHq.
diz tffiftl therefore AC AG :: DF. DH. whence by ad-
22.6. dkion AC -4- AG. AG :: DF ~\- DH. DH. that
e 22. i?, 2 AB. AG :: 2 DE.DH. e confequently AB.
f 1 7 c. AG :: DE. DH j / whence by di vifion AG .
GB.-DH. HE. Winch was to be dcm.

b $o. 6,

C47-I- I—, ; K

d 4. 2.

e 10. 13. m o d

f 1, J/;* /jrm* enrfe JBD comprehends both ABCDE

3. ax. the pentagone of a Vodecaedro7ty and LMNthe tiian-
£ 8. 1 5. £/e 0//772 Icofaedron infer ibed in the fame Jphere.
hi. ij.gf Draw the diameter AG. and the right lines
16. S* AC, CG. and let IK be the diameter of the
kzz.&gffphere, a and IKq = 5 OPq. b and make OP.

4. 4- OQ.y OQ. QP. Becaufe ACq -4- CGq c =3 AGq
1 151?. — 4 FGq 5 and ABq e r^FGq. f thence ACq
in conjlr. -+ABq= 5 FGq. moreover, becauie CA. AB^.v
n cor. 16. AB. CA — AB ; and OP. OQ.: OQ; QP. h and
15. fo CA.OP AB.OQ. k therefore 3 ACq (/IKq)

0 rz. 1;. 5 OPq (m IKq) 3 ABq. $ OQq. therefore 3
p to. 1 j. ABq s= 5 OQq. But becaufe ML n is the fide
cl 1 $• $• of a pentagone inferibed in a circle, whofe ra-
• before dius is OP, thence 1 5 RMq. 0 == S ^Lq t — 5
r 1. tf.v. 1, OPq 5 OQq= * 3 ACq-n 3 ABq $ = 15
andfeh. FGq. r therefore RM = FG. /and confequently
4^ 1. the circle ABD is — tg the circle LMN- Winch

1 i. def. iwasjo be dcm,

PROP*

EUCLIDE'/ KUmmtZ $41

PROP. IV,

a 8. 1.

If from F the center of a circle encompajjing the*
fentagone of a dodecaedron JBCDE, a perpendicu-
lar line FG be drawn to one fide of the fentagone CD;
the reft angle contained under the J aid fide CD and
the perpendicular FG, being thirty times taken, is
equal to the fuperficies of a Dodecaedyon. Alfo,

If from the center L of a circle inclofing the trian-,
gk of an Icofaedron HIK, a perpendicular line LM
be drawn to one fide of, the triangle HK„ the
reS angle contained under the faid HK, and the
perpendicular LM, being thirty times taken, Jball be
equal to the fuperficies of an Icofaedron.

Draw FA,FB,FC,FD,FE. a then {hall the tri-
angles CFD,DFE,EFA,AFB$FC be equal, but » 4-
CD x¥Gbz=z triangles CFD. therefore go c 15.
CDxGF<;= 60 CFD d = 11 pentagqnes a 0.
ABCDE e^= to the fuperficies of a dodecaedron. e 17. 3*
Which was to be dem. r 41. 1.

Draw LI, LH, LK; then HK x LM / is = 8 !•
2. triang.LHK. therefore 5oHKxLM^~ 60 11
HLK ■= 10 HIK h — to the fuperficies of an
Icofaedron. MHrich was to be dem.

Coroll.

CD x FG. HK x LM k :: the fuperficies of a /
dodecaedron to the fuperficies of an Icofae-
dron. *

Y j PRC"

/

/

/

/

/

34*

a j. 13,

b 9- 13.
c 1. 14.
d cor. 12.

e 15.
f for. 17.

'J*

g *• M-
h i- 6.

Jk 7. 5-
1 ^r.4.14

7ie fourtttnth Book of
PR OP. V.

Thefuperficiesofa Do-
decaedron has to the fuper-
ficies of an Icofaedron in-
fcribed in the fame ghere%
the fame frofortion that
H the fide of a cube has
to AD the fide of an Icq-
faedion.

_ Let the circle ABCD

a inclofe both the pentagone of a dodecaedron,
and the triangle of an Icofaedron 5 whole fades
are BD, AD. upon which from the center i. Jet
fall the perpendiculars EF, EGC, and draw

C Becaufe EC -] CD. EC b :: EC. CD. thence
EG Co ± EC - CD.) EF jM 4 EC) 1 j:: EF. EG
- EF. (*CD0 but H. BD/:: BD. H - BD, «
therefore . BD. :: EG. £F. confequently H *
EF BD x EG. wherefore fince H. AD a ::
H x EF. AD x EF. k thence fhall be H. AD ::
BD x EG. AD x EF / :: the fupetficies ot a do-
decaedron to the fuperficies ot an Icolaedrorn
Winch was to h dm. ¥

PROP,

EUCLIDE'* Elements*,

941

PJIOP. VI.

If a tight line AB be
cut in extreme and mean
proportion, then as the
right line BF,containing
in power that which is
made of the whole line
AB, and that which is
made of the greater feg-
ment AC, is to the right
line Econtainingin power
that which is made of the wMe line AB, and that
which is made of the lejfer fegment BC ; fo is the fide
of the cube BG to the fide of fin Icofaedron BK in-
ferred in the fame fphere with the cube. ^ •

In the circle, whore femidiameter is AB, in-
fcribe BFGHI the pentagone of a dodecaedron,
and BKL the triangle of an Icofaedron, a where- a cor.ij,
fore BG fhall be the fide of a cube inferibed in 15.
the fame fphere. therefore BKa b ~ 3 ABq; and b 12. 1 j
Eq c— $ ACq. therefore BKq. Eq<2:: ABq.c^ ij.
ACq e :: BGq. BFq, wherefore by inverfion BGq. d 1 j. j.
BKq BFq. Eq. / whence BG. BK BF. E. e i. 14
Which was to be dem.

f zi. 6.

PROP. VII.

♦

v
•

A Dodecaedron is to an Icofaedron, as the fide of
a Cube is to the fide of an icofaedron, inferibed in
one and the fame fphere.

Becaufe a the fame circle comprehends botl* a J» 14.
the pentagone of a dodecaedron, and the triangle
of an Icofaedron, b the perpendiculars drawn b 47. u
from the center of the fphere to the planes of
the pentagone and triangle, fhall be equal one to
another. Therefore if the Dodecaedron and Ico-
faedron be conceived divided into pyramided,
right lines being drawn from the center of the

Y 4 fphere

j{44 fourteenth Booh of, Szcl

Iphere to all the angles, the altitudes of all the
py ram ides fhall be equal one to the other,
c and6. Wherefore fince the pyraroides of equal heighth
ii. with the bafes, and the fuperficies or the dode-
caedron is equal to twelve pentagones, and the
fuperficicies of the Icofaedron to twenty trian-
gles, the dodecaedron fhall be to ^he Icofaedron,
as the fuperficies of the dodecaedron is to the
^5.14, fuperficies of the Icofaedron, d that i.% as the fide
of the cube is to the^fide of the Icofaedron.

>l -J -

PROP. VIII.

The fame circle
B CD E compre-

bends both the
fquate of the cube
BCDE } and the
triangle of the 08a-
edron FGH inferu.
bed in one and the
fame Jfyhere*

0 Let A be the diameter of the Iphere. Becaufe
ot<t-> Aq <i r=: 3 BCq b r= 6 BIq ; and alfo Aq c =2! z
b £' i*' GF(i d - <$ KFq h thence fhall BI be = KF. e
therefore the circle CBED = GFH* Which
d 21. 13". wa£ t0 be demonjlrated.
e 2. A/. 1,

The End of the fourteenth Book.

9 • %

THE

1 *

?4r.

THE FIFTEENTH BOOK
EUCLIDE j ELEMENTS.

PROP. L

• «

12^ 4 cube given JBGHDCFE to defcribc a
pyramide JGEC.
From the angle C draw the diameters
CA, CG, CE 5 and conneft them with
the diameters AG, GE, EA. All which
are a equal among themfelves, as being the dia- a ^ t
meters of equal fquares : therefore the triangles
CAG, CGE, CEA, EAG are equilateral and
equal , and confequently AGEC is a pyramide,
which infills upon the angles of the cube, and
therefore b is infcribed in it. Winch was ftbji,
I* done. JI#

PROP.

34*

Tie fifteenth Book of

PROP. li-

ft 10. I.

b 4. ii.

In a pyr amide given ABDC
to defcribe an o&aedron EG-
KIFH.

a Bifeft the fides of the
pyramide in the points E,
I, F, K, G, H, which join
with the right lines EF,
FG,GE,gfr. Allthefeare

b equal one to the other ; confequently the 8

triangles EHI, IHK, gjV. are equilateral and

C *7- def. equal, and fo make c an o&aedron described d

15. in the given pyramide. Which was to be done*
d 11. def.

PROP. Ill,

\ ; •

In a cube given CHGBDEPJ to defcribe an
oSaedron NP^SOR.
* 8. 4. Conned * the centers of the fquares N,P,Q,S,
0,R,with the twelve right lines NPtPQ,QS,©V.
a 4. 1. which are a equal among themfelves ; and fo
b 31. and nuke eight equilateral and equal triangles:
%l.def 1 1. wherefore b the Oftaedron NPQSOR b is in-
ferred in the cube. Winch was to be done.

PROP.

EUCLIDFi Elementi.

347

PROP. IV. * -

In Ociaedron given
AECDEFy to infcribe &
cube. $ *

Let the fides of the
pyramide EABCD,
whofe bafe is the fquare
ABCD, be equally bi-
fefted by the right lines
LM, MN, NO, OL,
which are a equal and a 4- *•
I parallel to the fides b 2- 6.
of the fquare ABCD. <?c zo^'fa-
then the quadrilateral
LMNO is a fquare. In like manner, if the
fides of the fquare LMNO be equally bifefted
in the points G, H, K, I, and GH, HK, KI, IQ
connefted, GHKI fliall be a fquare. And if in
the other 5 pyramides of the oftaedron, the cen- .
ters of the triangles be in the fame fort conjoin-
ed with right lines, then other fquares will
be defcribed like and equal to the fquare
GHKI. wherefore fix fuch fquares mall make a
cube, which lhall be defcribed within an ofta-
edron, d being its eight angles touch thedji.dtf.
eight bafes of the oftaedron in their centers. U«
Winch was to U don$*

PROP,

i by Google

348

The fifteenth Book of
PROP. V%

c p If

In an Icofaedron given to infcrile a Dodcc&edron.
Let ABCDEF be a pyramide of the Icofae-
dron, whofe bare is the pentagone ABCDE; and
the centers of the triangles G,H,I,K,L ; which
conneft with the right lines QH, HI, IK, KL,
LG. Then GHIKL fhall be a pentagone of the
dodecaedron to be infcribed.

For the light lines, FM, FN, FO, FP, FQ,
a coy.%. -.padiiig by the centers of the triangles, a do e-

b 4. r
c 4. i.
d 8.1.

e'4. 1.
f 11. 13.

qually divide their bafes into two parts, h there-
fore the right lines MN, NO, OP, PQ, QM c
are equal one to the other \ d whence alfo the
angles MFN, NFO, OFP, PFQ, QFM are e-
qual. therefore the. pentagone GHIKL is equi-
angular, e and confequently equilateral, being
FG, FH, FI, FK, FL / are equal. And if in
the other eleven pyramides of the. Icofaedron,
the centers of the triangles be in like fort
conjoined with right lines, then will penta-
cones, equal and like to the pentagone GHIKL,
be defcribed. Wherefore 12 of fuch pentagofles
fhall conftitute a dodecaedron ; which alfo
fhall be defcribed in the Icofaedron, feeing the
twenty angles of the dodecaedron confift upon
the centers of the twenty bafes of the Icofae-
dron. Whereby it appears that we have defcri-
bed a dodecaedron in an Icofaedron given.
Which was to he do^t

EUCLIDFs

i by Google

EVCLIDE's DATA.

Commentary or Preface written by
the Pbilofopher MARINUS, on
EUCLIDE'j DATA.

N the firlt place we ought to fet down
(as a Foundation) what that is, which
we call DATUM or GIVEN } then to
confider the Profit and Utility thereof 5
and in the third place, to what Art or
Science this Traft doth appertain.

The Word DATUM therefore is diverfly defi-
ned, for the Antients have defined it alter one
manner, and later Writers after another, whence
it follows that it feemeth a difficult thing to
give a true Explication thereof; for fome of
them have not delivered the Definition of the
Word ; but have with much Labour and Trou-
ble fought certain Proprieties thereof, and fome
others collecting and mingling what hath been

delivered

349

Digitized by Google

EUCLIDE'i DATA.

delivered by others before, have endeavored to
define the Word DATUM; but not fo exquifite-
ly but that they have contradicted themfelves ;
altho' what hath been laid by all of them,
feems to be grounded on one and the fame
notion and fuppoiition ; for they all take the
Word DATUM to be a thing comprifed \ and
therefore among fuch as have endeavored to
defcribe it molt limply, and with fome fimple
difference, fome of them have taken the Word
DATUM to be the fame with ORDINATUM,
and fo ApoUonm underilands it in his Tract
of Inclinations, and in his univerfal Trad: ;
and fo others, as Diodorvs takes it to be COG-
NITUM KNOWN ; for in this Signification
he takes the right line and the angles to be
given, and all that may arrive to our Know-
ledge, altho' we may not be able well to ex-
prels it. But others have believed that it
hath the fame fignification as the Word [Efa-
We] that may De declared, and fo Ptolomy
would have it, who calls thole things GIVEN,
whofe meafure is known whether precifely, ot
near the matter. Others alfo have thought the
Word DATUM to be what is granted us by
the Propofer in the Hypothecs ; being that
in the hrft Elements, a point given, and a
right line given, is diverily taken (that is to
fay, that who fo would give and determine
the quantity of a right line) all which things
fignify fome COMPREHENSION \ and there-
fore of all thefe Definitions, thole are moll
agreeable, which do molt openly declare the
COMPREHENSION, as we mall make evi-
dent by what follows.

Let us now unfold the diverfe Opinions of
thole, who writing the nature of DATVM
GIVEN, have not taken one iimple Mark, or
only Character for its Definition \ and let us
reduce it as in a Summary or tpitomy, to the

EUCLIDE'i DATA.

end we may with the more eafe know or num-
ber all their Differences. Some of them then
have defined DATUM to be Ordinatum and Pa-
rimon together, and others Otdinatum and Cog-
nitum together, and others Porimon and Cog-
?iitinn together. Wherefore all feem to have
io defined it as to have had regard to the
Conifrehevjio7iy or AJfuming and Invention of the
thing given ; and to the end that we may the
better conceive their Opinions, and that from
the faying of many we may be able to> draw
a true Definition of what is propofed, we will
take notice in the firft place of the Significa-
tion of all the fimple Terms which they make
ule of, as alfo of the Terms oppoled to them,
to wit, Inordinatum and Incogwtum, Jporon and
Irrational 5 for thofe things appertain to this
Geometrical Bufinefs, to natural things, and to
Mathematical Difcipline.

Now we may call that Oriinatum (or Regula-
ted) which doth always keep aud obferve that
for which it is faid to be ordered, whether you
regard its Magnitude or Species, or touching
fome other fuch like thing : It is alfo thus
defined, Chdinatum is that which cannot be done
in divers manners, but in one only manner, and
in fome determined ylace: As for example^
A right line drawn by two given points, is
faid to be ordered, by reafon it cannot be other-
wife done, nor in divers manners. But an an-
gle paiTing by two points is faid to be Inordina-
tum (or difordinate and irregular) for that it
is made in infinite and divers manners by a
great or fmall circle defcnbed by two points
ad infinitum, Contrariwife, an angle conlfcitu-
ted by three points, is faid to be Ordinatum,
as alfo thofe things which follow are faid to
be Ordinatum, as to conftitute an equilateral
triangle on a right line, for it cannot be di-
yerily made, but unchangeably, on both the

given

3fi EUCLIDE'* DJTA

extremities of the line. Again, To divide a
given Tight line according to a given propor-
tion, for that cannot be done but in one cer-
tain point. The things Inordinatum are fuch
as are* done contrary to thofe laft mentioned,
as to conftitlite a Scalene triangle, and to di-
vide a right line indefinitely. Wherefore the
Problem is ordered, is propofed in the deter-
mination, confidering that a certain thing may
be in one manner laid to be Ordinatum, and
Inordinatum in another, as an equilateral trian-
gle, confidering the equality ot the fides* it is
Ordinatum, but confidering its Magnitude it is
Inordinatum, being in no wife determined.

But we call that Cognition which is notorious,
as clear and comprehended of us, and Incogni*
turn that which is not known, or comprehend-
g ed of us, as the length of a way is faid to be
known, when we know how many Miles it
contains ; alfo that the three lines of a Refti-
line triangle are equal to two right angles ;
and in like manner that the Binomial is Irra-
tional, fuch things are known, as alfo that it
is only one right line that can touch a fpiral
line from a given point without it, from both
parts ; for if there were yet another line, two
right lines would enclofe a fpace, which is im-
poflible. Again, irrational things are not faid
to be unknown, but fuch of them only which
are neither known nor comprehended of us.

Porimon is that which we may make and con-
ftitute, that is to fay, bring to our Under-
ilanding. Again, ic is defined thus, Porimon
is that which may be exhibited by Demonftra-
tion, or which is apparent without Demonftra-
lion, as to defcribe a circle from a center aad
with a fpace, as alio to conftitute not only an
equilateral triangle, but alfo a Scalene ; or to
find a Bmomuim, or to find two. right lines ra-
tional, commenfurable in power only, and other
, things

>

m *

i by Google

EUCLIDE', DATA.

things which are known infinitely, are Forimon-
as to defcribe a circle by two points.

Apron is wholly oppofite to Porimon, as for
example, the Quadrature of a circle, for that
it hath not as yet been found ; altho' it be
certainly known that it may be: Neverthelefs?
the manner of finding it out hath not been to
this prefent compiehended. But we fpeak here
of that which is already known, which is
called Ponmon principale ; for what hath not
been as yet made, and yet n^ ^erthelefs is pofli-
blc, is called Porifton, (or feafible) altho' the
Conftruclion be yet unknown. But Jporon, as
hatli^ been afore faig\ is oppofite to Porimony
and is that whofe Nature is not as yet deci-
ded, nor well determined,

Effabile, that is to lay, rational (or fpeakable
and explicable) is that whofe Magnitude, Spe-
• cies, and Polition, we may be able to declare ;
but this Definition is a little too general, for
properly, md according to it felf, Effatute is
that which is known by certain thii.gs, and
according to a Meafure given by Pofition, as
of a fpan, or a finger's breadth, Qfa

Thefe things then being thus unfolded, we
rnay eafily perceive in what all thole things
that we have afore fpoken of do agree toge-
ther, and wherein they do differ ; and firfi: of
all how Ordmatum and Cogmtum do agree toge-
ther, and likewife their oppoiites the one to
the other, for it cannot be laid that any one
of thofe things by counterchanging is the
Other, nor yet that the one haih not more
extent than the other, altho' they agree in
many things, as to defcribe a right line by two
points, and to conftitute an equilateral triangle
by three -circles. But to fquare a circle, that
is indeed Ordhixtiim, yet neverthelefs, Incogm-
turn. Alio that at a point of a fpiral line
there is but one touch line, that is, of the

Z kind

EUCLIDE'j DATA.

kind we call Ordinatumy and cannot be other-
wife done, yet neverthelefs the Demonftration
and Conihuftiou thereof is not yet known.
i\gain, the indefinite feftion, and the Conftru-
ftion of the Scaienum is Ccgnnum ; but is not
Ordinatum ; infomuch as it is imnifeft, that
amongft thofe things which are Ordinatum 9
there are fome that are Cognitumy and others
that are Incognitum > and contrariwife, that
amongft the things that are Cognitum, there
are fome that are Ordinatum> and others that
are Inordinatum ; and therefore thofe things
anfwer one another, as among Living Crea-
tures, that which hath reafon, with that
which hath Feet, for there is no equality
amongft them, neither doth the one extend
more than the other.

In like manner, Ordinatum and Inordinatum
agree together, refpeding Potimo?i an&Jjtoroni
feeing that between them there is a very great
Refemblance, and becaufe that they do differ
only in the manner before expreiTed i for in
truth the fpiral line is Ordinatum ; but it was
not Porimon before Jrcbimedes \ and by the fame
reafon thofe things that are inordinate and
known by an infinity of ways and means are
Porimon, if any one fhall undertake to invent
their Conftnution and Conftruftion. Yet ne-
verthelefs they are not ordinate, as to confti-
tute a Scaienum triangle, it being no difficult
thing to make known the Conftnution thereof
by an equilateral triangle, yea it is moft eafy,
altho' it be inordinate and known by an infini-
ty of ways.

And in the fame manner do agree Ord'nut-
turn and Inordinatum, together with Efabile and
Irr&tionale ; for they agree together in many
things, differing neverthelefs by the foregoing
reafon, feeing thofe things there mentioned bear
110 equality to each other, neither doth one

thing

EUCLIDE'j DATA.

thing contain* the other ; for all Binomiums and
luch as are taken as Irrationals, are indeed
ordinate, but yet they are not Effabiie, or ex-
prcflible, or to be unfolded, as the diameter of
the f qua re is in refpeft of its fide. Now
touching Effabiie, there are divers inordinate 5
becaufe they are diverfly known, and indeter-
minately, for a Scaienum triangle may be mea-
lured by a defined and piopoled mealure, as
explicable, altho' it be inordinate.

Now it is eafy to fee the agreemeht that there
is between Cognitum and Porimon, but it is a
difficult thing to exprels or unfold their diffe-
rence ; forafmuch as in their Natures they
come fo near to one another, as that there
ieems to be an equality between them : Ne-
verthelefs, there will forn^ difference appear to
him that lhall confider It more ftrichy ^ for
let it be confented to that there can be only one
line that can touch a fpiral line in a certain
point, that is Cognituvi, yet notwithstanding
the Problem is not Porimon^ it being not as
yet comprehended ; fo as that all that which
is Cognitum , is not therefore Porimon. But ail
that which is Porimon is alfo Cognition, and
therefore Cognition appears to be of a greater
extent than Porimon^

Now Cognitum, Porimon, and Ejfabile , da
agree in fome certain things, and du differ in
other things by the lame reafon before alledged ;
for thoTe lines which are there called a ratio-
rials are in truth known } and yet nevenheiefs
are not Effhbile or. explicable. Contrariwife,
every number is indeed Effabiie, and yet every
number is not Cognitum. But Effabiie is al-
ways of its own nature expreiRble, altho' that
fome lengths may be now Effabiie, and at ano-
ther time not, if it be examined with fome
other according to one and the lame meafure.
But alfo that fame length is fometimes known,

EUCLIDE'* DATA*

and other times not , tho* they -wholly
agiee with one another. Now it is a difficult
thing to find fomething that fhall be Efgbile
and inccgnitum, for Cognitim feems to be of a

r eater extent than Effabile, and by thofe things
is maniteft that Porimon and Apron do differ
from RATIONAL pr EfaHle, and from IR-
RATIONAL, for of IRRATIONAL fome of
them may be Fonmon 5 but of RATIONAL
none of them can be Irrationals: and therefore
it is very eafy to perceive in what the before
exprefled things agree. Notwithftanding they
feem to agree together, in fuch fort as that
Porimon feems to be of a greater extent than
Effalnlc. —

Now by thefe things we may come to know
the difference of tho^e things that have been
before fpoken of, for in truth Effabile and Irra-
tionale are fo termed in refpedl of meafure,
which notwithftanding is not as yet arrived
to our Underftanding, feeing that fomething
that is rational, may be as yet unknown to us,
and in like manner may be rational, and yet
may never be comprehended fo to be. But Or*
dinatum and tnorfanatum is fo termed according
to it felf, and according to the proper nature,
of the thing on which we contemplate, altho*
it be not comprehended by us. As Archimedes
had perceived fome things to be ordinate from
the nature of the things, the which Sereniu
had before contemplated. But Cognifum and
Incognitum is fpoken in refpeft of us, fo as
the things before mentioned do differ among
themfelves ; for thefe have refpeft to us, the
others, fome of them to their proper nature,
and the reft to meafure.

Having then explained the agreements and
differences of the things that have been pro-
pofed, it remains now that we confider what
is meant by the Word DATUM, for of all
♦ thofe

s

Digitized by

EUCLIDE'/ DATA

thofe that believe the Word DATUM to be
that which is confented to by the Propofer in
the Hypothefis, are wide from what is iought ;
becaufe that all the Elements of the things
GIVEN are not compofed of this fort tf
GIVEN, which is according to the Hypothe-
cs, as may be feen in thofe Tra&s which hav$
been made on this Suhjeft GIVEN. Where-
fore waving this Opinion, let us judge of the
Definitions of others.

Then, that which is confented to, or grant-
ed in the Hypothefis, is fomething which is
confequently known by the Principles ; but
fuch as make ufe of Definitions ot one only
Word, do define it and remark it by fome one
of the before mentioned, as hath been fpoken
in the beginning, fo as that almoft all feem
to have had this common notion of GIVEN,
to wit, that it is comprehended even as t he
Word DATUM, doth alfo manifeft it to be ;
aind amongft thofe, thefe are the chief that do
define it by the Hypothefis or Suppofition ; and
others have had regard to what is confented
to or granted. But we making ufe of the faid
things as of a Rule and Dire&ion to judge
'lightly, we may be able to find dut a petfeft
Definition of f)ATUM\ for it is certain that
it ought to equal and be convertible with the
thing defined, which is one thing proper to
good Definitions. Now fuch feems to be the
Definition of the thing propofed, which among
the moft fimple and plain Expofitors is defi-
ned PorimoV) and amongft the more acute, that
which defineth it to be Porimon and Cognitum
together ; but all the reft are imperfedt ; for
that which defineth it Or&inatum is not fufti-
cient for the Comprehenfion and Knowledge of
DATUM '\ becaufe that neither wholly ordi-
nate nor alone ordinate, is not comprifed, fee-
ing that there are things inordinate that have

EUCLIDE'f DATA.

the fame Condition, as hath been ihewn,.
Again, that reafon gives not Sathfadio%nei-
ther, which describes it to be Cognitwn • for
all that is known is not comprehended, altho'
that alone Cognitum be comprehended. More-
over, that alio is not perfect which defineth it
to be Effabile, for Efabile is not alone com-
prehended ; feeing that fome of the irrationals
are alfo comprehended. In like manner, all
Efabile is not comprehended, as hath been be-
fore declared. Now amongft the Definitions
which expound it by the only Word, there re-
mains that which defineth it to be Porimon,
which feemeth greatly to manifeft the Com-
prehenfion , for whole Porimon and alone Pori-
mon is comprehended. Wherefore EUCLIDE
himfelfufeih fuch ^t)efinition in a DeCcription
of all the kinds of GIVEN by him conceived
and regarded. But amongft fuch De6nitions
as are compounded, that is a perfect Definition
which defineth DATUM to De Cognitum and
Porimon together, having Cognitum for analogi-
cal kind, and Porimon for difference 5 but that
is imperfect which hath Ordinatum and Porimon
together, for thofe things which are fuch, are
not alone GIVEN, and that which defineth W
Ordinatum and Efabile together, comprehendeth
likewife the GIVEN, with the defect or want.
But that of Cognitum and Ordinatum together,
is not to be received or admitted, becaule it
doth exceed what is defined, for fuch is not
given alone : Therefore thofe only which have
declared that DATUM is Cognitum and Porimon
together, feem to have attained the notion of
GIVEN, for that which is fuch is all, and
alone comprehended, which two things ought
to be in thofe Definitions that are well given.
But the former comes near to thofe which
have thus defined it : DATUM is that to which
we^may find an equal, according to thofe things

'we

A

EUCLIDEV

we have propofed in the firft Principles anc|
Hynpthefis, of which number EUCLIDE is one,
making ufe throughout of the Word metnida.*
which fignifies to exhibit or invent, altho' he
leaves Cogkitum as a Confequent of Ponmon ;
fome one neverthelefs might reprove him, for
that in the 6rft place he hath not defined DJ-
JVM. in general : but immediately fome or the
kinds of GIVEN, altho' in his Elements of
GEOMETRY, he hath defined the line hmple
before the Species or Kinds.

What is the Utility and Profit that
arifetb from this TraB of DATA,
. or things GIVEN,

Fter having explaine£<univerfally, and ac-
/ » cording to what feemed neceffary for our
prefentUfe, what this Word D^r&jKiignifieth:
It follows to ihew the Utility of this 1 raft.
Now this Traft is fuch, as that it is not only
ordained and inftituted for its own refpeft, but
for fome other thing; for it is very neceffary
to a place which is called Refolved, and we
have already declared elfewhere how inuch
^Strength a refolved place doth obtain in Ma-
thematical Difciplines, as alio in Opt.cks ; and
Cannons, which come very near to them, , as
well for that Refolve is an Invention ot the
Demonftration, as for that in fuch like things
it [erves us much for the Invention ot the
Demonftration, or for that it is much more ex-
cellent to meet with a Refolutive Power, than
to enjoy divers particular Pemonlbations.

To what Art or Science this Tratl

is referred.

OW feeing the Confideration of GIVEN
is ufeful and profitable in all thefe kinds

Z> 4

N

EUCLIDE'* DATA.

of Arts, for that it ferveth much to RESOLU-
TION, it may well be faid to be recalled, not
only to one only Science, but to the Mathe-
matics universally which treat of Numbers,
Time, Swiftnefs, and fuch like things, which
treateth hkewife of Reafons, as alfo of Pro-
portions, and in a word of all Medietez : Where-
fore for the perfeft and demonftrative Know-
ledge of things GIVEN, being of fo great
Utility, EICLIDE hath taken pains to frame
this Book of things GIVEN, which Author
amongft all fuch as have compofed the Ele-
ments of Geometry, hath juffcly deferred the
firft place and rank, and who having invented
the Elements, or rather the Introductions a>
inoft of all Mathematical Difciplines, to wit,
of all Geometry in Books, of Aftronomy,
of Mufick, and Opticks, he hath left in Wri-
ting the Elements RESOLUTIVE, in this
Treaiife of things GIVEN ; but as he was a
Geometrician, he hath particularly accommoda-
ted to Magnitudes, that was of the GIVEN;
yet nevertheless common in other things, which
Method hath alfo been obferved by him, when
treating univerfally of Reafons and Proportions,
he appropriates them to the Magnitudes meo-i*
tioned in his Fifth Book of Planes.

Now it hath been declared in general what
is the meaning of DATUM, to what Science
it appertained, and how profitable the Con-
templation thereof is. We will add to what
hath been faid, the Defcription of this Sci-
ence which treats of things GIVEN ; feeing
that it is (as appears by what hath been faid)
a Comprehenfion in all manners, of things
GIVEN $ and of their Accidents and Proprie-
ties. But having refpeft to the propofed Book,
we fhall declare it to be an Elementary Do-
ctrine of the whole Knowledge of things
GIVEN, whence it follows that it will be

very

Now this Book is divided according to the
Species or Kinds of the things GIVEN, and
in the firft Section are contained thofe things
which are given by Reafon. Secondly, fuch
as are given by Pofition : and laftly, fuch as
are given by Species or Kind ; for that which
is given by Magnitude is fimple, and particu-
larly contained in the others, and principally
in thofe things given by Species or Kind. Now
he hath begun with thofe things given by
Reafon and Pofition, forafmuch as thofe that
are given by Species are cpnftituted of them.
EUCL1DE gives yet another Divifion to this
Book, for that he divides it into univerfal
Magnitudes, Lines, and Superficies, and into
circular Theoremes, which Order he hath alfo
fcbferved in the Definitions and Suppofitions of
this Book. Moreover, he ufeth a certain way
of inftrufting, which proceeds not by Com-

The End of the PREFACE

MARINUS.

. DEFINITIONS.

Lanes or Spaces, Lines, and Angles, tQ
which we may find others equal, are
[aid to be given by Magnitude.
II. J Reafon is f aid to be given, when
we may find one of the fame or
equal thereto.

III. ReSiline figures, fhofe angles are given, and
alfo the uafonofihe fides to one another, are
[aid to be given by Species or Kind.

IV. Points, Lines, and Angles, which have and
keep always one and the fame place and fituation,
art faid to be given by Pofition or Situation.

V. A Circle is faid to be given by Magnitude,
when the femidiameter thereof is given by Mag-
nitude.

VI* A Circle is faid to be given by Tofition, and
by Magnitude when the center thereof is given*
by Pofition, and the femidiameter by Magnir
tude.

VII. Segments of Circles, whofe angles and bafes
are given by Magnitude, are faid to be given
by Magnitude.
Vlil. Segments of a Circle, whofe angles are given
ly Magnitude, and the bafes of the fegments by
Pofition and Magnitude, are faid to be given,
f by Pofition and by Magnitude.

IX. A Magnitude AB, is
D greater than another

< — - - ■■ . ■ B Magnitude C, by a given
C — — m Magnitude BD, wbtn

having taken aw ay. tic

given

Digitized by

EUCLIDE'x DATA.

given Magnitude DB> tie rejl AD9 is equal to
the other Magnitude C.

X. A Magnitude ABy is lefs than another Magni-

tude C, by a given Magni-
B tude BD .when ha vivg 'a dded

A — D thereto the given Magnitude

C » BD, tie whole AD is equal

to the other magnitude C

XI. A Magnitude AB, is [aid to be greater thar^

another magnitude
D C C£, by a given

A « ' B magnitude AD, andr

in reafon, when ta-
king from the fame magnitude the given mag-
nitude AD, the refi DB> hath to the other mag.
nitude CBj a given reafon.

XII. A magnitude AB is faid to be lefs than ano?

tier magnitude BCL
A B by a given magni-
fy _ C tude AD, and in

reafon, when the
given magnitude AD being added thereto, the
'whole DB hath to the other magnitude BC> a
given reafon,
XIIL A right line is faid to be drawn down from
a given joint, unto a right line given in Por-
tion, the right line being drawn in a given
angle.

XI V. A right line is faid to be drawn uf from a
given point, to aright line given in Popion, the
right line being drawn in a given angle.

XV. A right line is againjt another right line in
Pofition, when it is drawn parallel thereto through
a given pint*

PRo?a

^4

EUCLIDE'* DATA,

PROPOSITION I.

a i. def.

b 7- 5-
c 16. 5.
d z. def.

a 2.

b 14.
c It def

I

1

A B CD

TJTO magnitudes J and B$
being given, the tea/on
they have to one another A to B9
u alfo given.

Bemonftration. For feqng that
the magnitude A is given, a we
may find one equal thereto, which let be C.
Again, forafmuch as the magnitude B is given,
we may alfo find one equal to that, and let
that be D. Therefore feeing that A is equal
to C, and B to P, as A is to C, b fo is B to D,
and by permutation, c as A ihall be to B, fo C
lhall be to D. Therefore d the reafori of A to
B is given, for it* the fame reafon as of C
to D, as we have found, and which ought to
be demonftrated.

PROP. II.

* »

If a given magnitude J, hath to fome other
magnitude B, a given reafon, that other' magni-
tude Bf is alfo given by magnitude.

Demonftr, For feeing that A is
given, we may find one equal
thereto, which let be C: And tor-
afrjmch as the reafon of A to B,

A B C D *s atf° given> wc may find*

one of the fame. Let it be
found, and let the reafon be of C to D. Now
feeing that as A is to B, fo C is to D ;
and by permutation, as A is to C, fo B is to
D : But A is equ^l to C, therefore b B fhall be
alfo equal to D. Therefore c the magnitude B
is given, feeing that thereto there hath been
found one equal, to wit, D.

PROP.

Digitized by Google

EUCLIDE'i DATA.

A

!

IB
I

tbt

PROP. III.

If given magnitudes AB and BCt
are compounded^ that magnitude
hat is compounded of tlxmjbatl be aljo
vvin*

Demonftr. For feeing that AB is
1 \ givefo, we may find one equal to it,
C F which let be DE. Again, feeing
that BC is given we may alfo find
one equal to that, which let be EF. Where-
fore feeing that DE is equal to AB, and EF
is equal to BC, the whole AC a is equal to a z* ax. li
the whple DF. Therefore AC is given, feeing
that DF is propofed equal thereto*

A D

B £

PROP. I||

If from a given magnitude A B9
there be taken away a given magnitude
JCy the remaining magnitude CB is
alfo given.

henwnftr. Forafmuch as AB is
given, we may find one equal
thereto, which let be DE. Again,
feeing that AC is given, we may
alfo find one equal to it, which let be DF. See-
ing then that the magnitude AB is equal to
the magnitude DE, arrd the magnitude AC to
the magnitude DF, the reft CB a ihall be a i*
equal to the reft FE. Wherefore CB is given,
for to it there hath been found an equal, to
wit, FE,

PROP- V.

If a magnitude ABy hath a given
Reafon to fame fart thereof AC% it
mil have alfo a given reafon to the

II fart remaining CJf.
Demonftr. Let DE be expofed as a
: given magnitude, and feeing that
BE the reafon of the magnitude AB,

tt

A D

lc If

Digitized by Google

?66 EUCLIDE'i DATA.

a z. def. to the magnitude AC, is given, a we may find
one of the Tame, which let be DE to DF ;
therefore the reafon of the fame DE to DF is
bz. pop. given, .But DE being given, fo is b alfo its
c 4. prop, part DF ; and conlequemly, c the reft FEn:
di. prop* Therefore d feeing that DE andfE are given,
the reafon of the fame DE to FE, is alfo
given. And foraffhuch as DE is to DF, as
A B is to AC, and by cohverfion, as DE to FE*
to.$: fo AB is to CB. But the reafon of DE to FE is
given, as hath been demonftrated , therefore
. the reafon of AB to CB is alfo given,

• w

Scholium. . .
From this it is evident that if a magnitude bath
to fome fart thereof a given reafon, by divifion*
the reafon that om fart hath to the other , fball
he ilfo given. For feeing that as DE is to FR9
fo is AB to. CB j by division, as DF to FE, fo AC
to CB. But it hath been demonftrated that the
parts DF and FE are given, and confidently their
reafon is alfo given : In like manner, therefore
the reafon of AC to CB is given*

ui

PROP. VI.

AD If two magnitudes AB and BC, ha-

ving to one another a given reafon, are
E compounded, the magnitude AC comr
founded of them, /ball alto have a given
reafon to each of them AB and SC. .
Dmovftr. Let the given magni-
C F tude DE be expofed, and feeing
that the reafon of AB to BC is
give'n ; let there be made one and the fame of
the faid DE to EF ; therefore the reafon of the
4 z.prop* fame DE to EF is given ; and therefore a the
magnitude DE being given, both the one and
the other of them DE and FE, is given,
b uprop. Wherefore b the vwhole DF fliall be alfo fiivefi.

There-

Digitized by

EUCLIDE'* DATA, tf7

Therefore c the reafon of the faine DF to eachc i.pof*

of them DE and EF, lhall be given. And for-

afmuch then as AB is to BC, lb is DE to EF ;

in compouading, d as AC is to BC, fo is DFd 18. $.

to EF : Therefore by conversion, as AC to AB,

fo is DF to DE. Therefore as the whole DF

is to each of the other magnitudes DE and

EF, fo the whole AC is to each of the magni-

tildes AB and BC : Therefore e the reaton ofe z.def.

the fame AC to each of the magnitudes AB1,

and BC is given.

p k 6 P. vif.

6 If a given magnitude AB,

& _____ — B he divided according to a given

reafon ACMo CBy each fig-
ment AC and CB is given.

Demonfir. For feeing the reafon of AC to
CB is given, the realon of a AB to each of
them (AC and CB) is alfo given. But AB is a 6- W
Given : Therefore b each o£ the fegments AC h fc
and CB is alfo given. 5 *•*"*-

. ; prop. vitr.

■

Magnitudes A and C>
which have to one and
the fame a given reafon
By fbatt he to one ano-
ther in a given reafon,
A to C.

Demonfir, For let the
given magnitude 0 be expofed, and feeing that .
the reafon of A to B is given, let the fame be
done of the faid D to E. Now feeing that
D is given, a E is alfo given. Again, feeing a z. fr6p*
that the reafon of B to C is given, let the fame
be done of E to F. But E is given, and there-
fore F is alfo given. But feeing that D is

given,

A

D

B

F

C

Digitized by Google

i. 1

B

£

C

F

;<58 EUCLIDE'* DATA.

t> x.pof* given, & the reafon of the fame D to F is given >
and feeing that as A to B, fo D to E, and as

c !%• $• B to C, fo is E to F 5 in reafon of equality, e as
A is to C, fo is D to F ; but the reafon of u
to F is given. Therefore the reafon of A to C
. is alfo given.

fckop- it

A D If two or niore m*g -

nitudes A, By and C, are
to one another in a given
reafon, : and that the
fame magnitudes A, B±
and C 9 have to * other
magnitudes D, £, and
F, Igiven reafons, fltho' they he not the fame, thofe
other magnitudes D, E, ana F, Jball he alfo to one
Another in given reafom.

Demonftt. Forafmuch as the reafon of A to B
is given, as alfo that of A to D, the reafon of
D to B lhall be given : But the reafon of B to
E is alfo given ; therefore the reafon of the fame
P to E lhall be in like manner given. Again,
feeing that the reafon of B to C is given,
and alfo that of B to E, the reafon of E to G
lhall be given. But the reafon of C to F is
a Z.frop. alfo given. Therefore a the reafon of E to F
fhall be given. But it hath bfcen demonftra^ed
that the reafon of D to E is alfo given ; and
b S.prof. therefore I the reafon bf D to F (hall be given;

Therefore the magnitudes D, E, and F, are to
one another in given reafoos..

PROP. X.

D B If a magnitude

A — — ■■■ - C JBy be greater than

another magnitude
EC, by a given magnitude, and in reafon, themag-

jiituai

Digitized by

EUCLIDE'* DATA. 369

intuit AC compounded of loth, jball he alfo gieater
than that fame magnitude, by a given magnitude,
and in reafon : But if that compounded Magnitude
he greater than the fame magnitude, by a given
magnitude, and in reafon \ either the remainder
jball he alfo greater than that fame hy a given
magnitude, and in reafon ; or elfe the fame re-
mainder is given with the following, to which thtt
other magnitude hath a given reafon.

Demonjlr. For feeing that AB is greater ttiari
BC by a given magnitude, and in reafon, let
the given magnitude AD be taken away.
Therefore a the reafon of the remainder DE^it.def.
to BC is given ; and in compounding, £ theb 6*P*o(*
reafon of DC to BC is alfo given. But the
magnitude AD is alfo given 5 therefore AC is
greater than the fame BC by a given magni-
tude, and in reafon.

Again, Let the magnitude AC be greater
than the magnitude BC, by a given magnitude,

and in reafon:
D\ B £ I fay that the

A C reft AB, is ei-
ther greater

than the fame BC by a given magnitude and
in reafon, or that the fame AB, with that
which followeth, to which BC hath a given
reafon, is given.

Forafmuch as the magnitude AC is greater
than the magnitude BC, by a given magnitude,
and in reafon, cutoff from it the given magni-
tude : Now the fame given magnitude is either
lefs than the magnitude AB, or greater : Let
it in the firft place be lefs, and let it be AD.
Therefore the reafon of the remainder DC to
CB is given. Wherefore by divifion, the rea-
fon of I)B to BC is given. But the magnitude
AD is alfo given ^ therefore the magnitude
AB is greater c than the magnitude BC by zciudef
given magnitude, and in reafon. Now let the

A a given

Digitized by Google

EUCLIDE'i DATA.

given magnitude be greater than the magni- •
tude AB, and let AE be put equal theretb j
A n. def. therefore d the reafon of the remainder EC to
e K.prov. CB Is given $ and by converiion, e the reafort
of the fame BC to BE, is alfo given- But the
fame EB with BA is given^ for that the whote
AE is given : Therefore there is given AB,
with that which follows BE, to which BC
feath a given reafon.

PROP. XL
E D B If a magnitude 0^

X- 1 — 1 — - -1 C be greater than a mag-
nitude BC, by a given
magnitude, a fid ifi reafon, the fame magnitude AB,
Jball he alfo greater than the magnitude compounded
of them by a given magnitude, arid in reafon, and
' if the fame magnitude be greater than the two others
together by a given magnitude, and in reafon, that
fame magnitude Jb all be alfo greater than the reft by
a given magnitude, and in reafon.

Demonftr. For feeing that the magnitude AB
is greater than BC by a given magnitude, and
in reafon ; let there be taken from it a given
a u.def magnitude AD : Therefore a the reafon of th€
b 6. prop, reft DB to DC, is given, and therefore b the
reafon of DC to BD fhall be alfo given : Let
the fame be done of AD to DE, therefore the
reafon of the fame AD to DE is given. But
c i. prop. AD is given, therefore c DE is alfo given,*
d 4. prop, and confequently, d the reft AE, is alfo given.

But feeing that as AD is to DE, fo is DC' ttf
e 16. 5. BD ; by permutation, e as AD is to DC, fo is
f 18. DE to DB : Therefore by compounding, /as
AC is to CD, fo is EB to DB ; and by permu-
P16.5. tation, £ as AC is to EB, fo is DC to DB, But
the reafon of DC to DB is given : Therefore alfo
is AC to EB, and confequently, that of EB to
AC. But it hath been demonftrated that AE is
h ii.def. given, therefore h AB is greater than AC by a
given magnitude, and in reafon.

Bat

Digitized by Google

EUCLIDE'i DATA. %11

But now let AB be greater than AC by a
iveh magnitude, and in reafon : I fay that the
Une AB is alfo greater than the reft BC, by
a given magnitude, and in reafori.

For teeing that AB is greater than AC by a '
given magnitude, and in reafon. Let the given
ifcagnitude AB be cut off there-frorfi : There-
fore i the reafon of the remainder EB to AC \%\ II%
given, and cbnfequently, alfo fhall be giverv
that of AC to EB. Let the fame be done of
AD to DE, therefore the reafon of AD to DE<
is given \ and by cohverfion, k the reafon of k f.prop*
AD to AE fhall be alfo given, and confequent-
ly that of AE to AD. Now AE is given, ,
therefoie the whole AD / fhall be alfo given >1 z.prop.
and feeing that as the whole AC is to the
whole EB, lo the part cut off AD, is to the
part cut off ED, fo alfo fhall be m the remain- m rp. 5,
der DC to the remainder DB. But the reafon
of AC to EB is given : Therefore alfo fhall
be given that of DC to DB. Wherefore by
divifion, n the reafon of BC to DB is given *nJcboh J-
and consequently alfo fhall be given that otP°l*
DB to BC. cut it hath been demon ftrated
that AD is given : Therefore 0 AB is greater o 11 • def.
than the fame BC by a giveni magnitude, and
in reafon.

PROP. XII.

B C If there he three mag-

A 1 1 D nitudes AB, BC, and

CD, and thai the fi>#
AB, with the fecond BC, to wit AC, be given.
But the fecond BC, with the third CD, to wit, BD,
he alfo given : Either thefirft AB Jball be equal to
He third CD, or the one Jball be greater than the
bther by a given magnitude.

Demohftr. Foraimuch as each of the magni-
tudes AC and BD are given, the given magni-

A a z tudes

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372 • EUCLIDE'* DATA.

tudes are either equal to one another, or une-
qual. Let them be firft equal : Therefore AC
is equal to BD, take away the common mag-
a J. ax. r. nitude BC, and there will remain a AB, equal
to CD. Butfuppofe them to be unequal, as
in this fecond figure, and let BD be greater

than AC : Let then
B %C E BE be put equal to

A — 1 1 — 1 D AC. Now feeing

, <. that AC is given*
BE is alfo given. But the whole BD is alfo
b £ pop given, the reft^ ED b fhall be fo, alfo ; and
torafm

Lmuch as BE is equal to AC, taking away
c 3. ax, lithe common magnitude c BC, there will re-
main AB equal to CE. But ED is given :
Therefore CD is greater than AB by the given
magnitude ED.

Scholium.

And if the firft with the fecond, to wit, AC% wert
greater than the fecond with the third, to wit, BD9

as in the other
E B C figure, CE would

A - — 1 - ^ — 1„ -l— D be made equal to

the fame BD> and
by the fame reafons as was above demonftrated, that
AE is given and equal to CD ; and therefore AB
greater than CD by a given magnitude.

PROP. XIII.

H If there le three magiiU

A 1 B iudes AB, CD, and E, and

F that the firft of them AB,

C 1 D bath a given reafon to the

E fecond CD ; but the fecond

■ * i . CD is greater than the third

E, by a given magnitude t
and in reafon, alfo the firft AB, fiall be greater

than

Digitized by Google

EUCLIDE'x DATA 37|

than the third E, ly a given magnitude^ and in
reafon.

Demonftr. For feeing that CD is greater than
E by a given magnitude, and in reafon, let
the given magnitude CF be taken there-from :
Therefore the reafon pf the reft FD to E is
given. And forafmuqh as the reafon of AB
to CD is given, let the fame be done of AH
to CF, Therefore the reafon of the fame AH
to CF is given. But CF is given : Therefore
a AH is alfo given. And feeing that as the a z.pop,
whole AB is to the whole CD, fo the part
cut off AH is to the part cut off CF, and fo
I alfo the reft HB is to rhe reft FD, the rea- b 19, $•
fon of the fame HB to FD is alfo given. But
the reafon of FD to E is alfo given : There-
fore c the reafon of HB to E is given. But it c 8.prop9
hath been deiponftrated that AH is given :
Therefore d AB is greater than the faid £ by d ir, def.
a given magnitude, and in reafon,

PROP. XIV.

B G If two tuagvir

A 1 1 E tudes AB and CD,

D have to one anothet

C 1 — F a given rcafon.and

that to each of
them there le added a given magnitude, to wit,
VE and DF \ either the whole AE and CF JbaU
have to one another a given teafon, or the one
fball le greater than the other by a given mag.
nitude, and in reafon. '

.Demonftr. For feeing that each of thofe mag*"
nitudes BE and DF, is given, a the feafon of a i,prof.
the faid BE and DF is alfo given 5 and if that
reafon be the fame with that of AB to CD,
that of the whole A£ to the whole CF, I b n» J,
lhall be the fame ; and therefore the reafon of
the faid AE to CF is given.

A a 1 NoV

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g74 EUCLIDE'j DATA.

Now let the reafon of BE to DF be not the
fame, with that of AB to CD, and let it be
as AB to CD, fo BG to DF, Therefore the
reafon of the faid BG to DF is given. But

€ z.frop. the magnitude DF is given, therefore c BG
> is alfo given • and feeing that the whole BE

44-/wp. is given, i the reft GE fhall be alfo given.
But forafmuch as AB is to CD, as BG is to

e 12. 5, DF, efo alfo is' the whole AG to the whole
CF , and therefore the reafon of the faid AG
to CF is given : But tke magnitude GE is

f rr. def. ^given : Therefore / the magnitude AE is5

' greater than the magnitude CF by a given

magnitude, and in reafon. ;

PROP. XV.

G If two magnitudes

1 B AE and CD, have to

one another a given
-D reafon', and that from
each of them be taken
away a given magnitude (to wit, 'from the magni-
tude AB the magnitude AE, and from the magni-
tude CD the magnitude CF) the remaining magni-
tudes EB and FD, either fbaU have to one ano-
ther, a given reafon, or the one of them Jball be
greater thdn the other by a given magnitude, and
hn reafon.

Dcmonftr.YQT feeing that each magnitude AI
and CF is given, the reafon of AE to CF ii
given ; and if it be the fame with that of
AB to CD, that of the remainder EB to the
a 19. 5. remainder FD, a fhall be alfo the fame 5 and
therefore the reafon of the faid EB to FD

fliall be alfo given.

E , Q But if it be not the

A 1 — — i B fame, let it be as AB

F to CD, To AC tp CF:

C 1~ -D Now the reafon of

~ AB to CD is' given;

there-

• :

Digitized by

EUCLIDFj DATA 375

therefore alfo that of AG to CF fliall be given.

But CF is given, therefore b AG is given.b i.prop.

But AE is alfo given, therefore c the reft EGc 4.frof,

is given \ and feeing that as AB is to CD, fp

the part cut off AG is to ttje part cut off CF,

and fo alfo is d the reft GB to the reft FD ; thed 4. prop.

reafon of the faid GB to FD is alfo given.

Therefore feeing that EG is given, EB is

greater than FD c by a given magnitude, ande luief,

in reafop,

PRO^. XVI.

J two magnitudes AB
CD, have to one
another a given reafon
D and that from one of
them, to wit, CD, ther*
te taken away a given magnitude DE, and to
the other AB there he added a given magnitude
BF9 the whole AF Jball be greater than the rejt
C£, hy a given magnitude, and in reafon.

Demoitfr. For feeing that the reafon of AB.
to CD is given, let the fame be made of BG
to DE : Therefore a the reafon of the faid BG a 2. aef.
to DE is given. But DE is given, therefore
I BG i &o given. But BF is alfo given, b z.frop
therefore c the^whole GF is given. And(ee-cj./^
ing that as AB is to CD, fo the part cut off
BG, is to the part cut off DE 5 and <i fo alfod 19. 5.
is the remainder AG to tlie remainder Cfc ;
the reafon of the faid AG to CE is given :
But GFis given, therefore the magnitude A*
is greater than the magnitude CET>y a given
magnitude, and in reafon.

Digitized by

37<S EUCL1DE/ DAT J.

p

PROP. XVII.

F If there he three mag-

A 1 B mtudes AB, E, and CD,

. w E and that the firjl AB he

G 11 greater than the fecond

C' ■ l>- D jb, by a given magnitude i

and in reafon. But the
third CD be alfo greater than the fame fecond E,
by a given magnitude, and in reafon ; the firft AB
jhall have to the third either a given reafon,
or elfe the one JbaU be grmh than the other by a
given magnitude, and in reafon.

Demovfir. For feeing that AB is greater thai*
E by a given magnitude, and in reafon, let
the magnitude AF be taken away : Therefore
the reaion of the* remainder FB to £ is given*
Again, feeing that CD is greater than the
faid E % a given magnitude, and in reafonr
let the given magnitude CG be cut off there-
from y and the reaion of the remainder GD to
Ztfrop* E lhall be given : Therefore a the reaforp of FB
to GD fhall be alfo given. But to the faid
FB and GD are added the given magnitudes
AF and CG : Therefore the whole AB and CD
b 14.jp> of ib lhall either have to one another a given rea-
fon, or the one lhall be greater than the other
by a given magnitude, and in reafon.

PROP. XVIII.

> * • •

B If there be three mag-
Ik 1 H mtudes AB, CD, ahi

I G EE, and that the one

C > D of them, to wit, CD, be

F greater than either of the

E li K other AB or EP, by a

<£>.■ given magnitude, and in\

reafon $ either the two others AB and EF9 JbaU

have

EUCLIDE'* DATA. iTJ

have to one another a given reafon, or the one jball
he neater than the other by a given magnitude,
ana in reafon.

Demonftr. Forafmuch as the magnitude CD
is greater than the magnitude AB Dy a given
magnitude, and in realon, let the given magni-
tude DG be taken there-from : Therefore the
reafon of the remainder CG to AB is given.
Let the fame be made of GD to BH, there-
fore the reafon of the faid DG to BH is given.
But DG is given, therefore a BHis alio given, a 2* pro*.
And feeing that as CG is to AB, fo is GD to
BH, b fo alio is the whole CD to the whole b 12. $.
AH, the reafon of the faid CD to AH fhall
bealfo given.

Again, feeing that the fame CD is greater
than EF by a given magnitude, and in reafon ;
let the magnitude DI be cut oft there-from :
Therefore the reafon of the remainder CI to
EF is given : Let the fame be made of DI to
FK. Therefore the reafon of the faid DI to
FK fhall be alfo given. But DI is given,
therefore FK is alfo given. And feeing that
as CI is to EF, fo is ID to FK ; fo alfo is the
whole c CD to the whole EK ; the reafon of c 5-
the faid CD to EK fhall be given. But the,
reafon of the fame CD to AH is alfo given :
Therefore d the reafon of the faid AH ro EK ■ 8. prop.
fhall be given. And feeing that from the
faid AH and EK, the given magnitudes BH
and F K are cut off, the magnitudes AB and
EF e are either in a given reafon to one ano-e l$'p*op*
ther, or the one is greater than the other
by a given magnitude, and in reafon.

PROP,

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578 EUCLIDE's DATA.

PROP. XIX.

If there he three mag-
B nitudes JBy CD, and E,
and that the firfi AB, he
P greater than the fecond
CD, hy a given magni-
tude, and inSeafon^ and
that the fecond CD he
greater than the third hy a given magnitude,
and in reafon ; alfo thefirjl magnitude JB fiall he
greater than the third E, hy a given magnitude, and
m reafon.

Demonjlr. For feeing that CD is greater than
I by a given magnitude, and in reafon ; let
the given magnitude CF be taken there-from :
Therefore the reafon of the remainder FD to
$ is giVeq. Again, feeing that AB is greater
than the fame CD by a given magnitude, and
in reafon : Let the magnitude AG be taken
^ihere from : Therefore the reafon of the i^main-
der GB to CD is given : Let the fame be made
of GH to CF : Therefore the reafon of the
faid GH to CF is given. But CF is given :
Therefore alfo GfJ is given, and then AG isK
j./rop, ' _ 1 alfo given, the whole a

AH fliall be alfo given.
But as QB is to CD,
fo is GH to CF, and
C 1 -D fo alfo* the

HB to the remainder
FD : Therefore the rea-
fon of the faid HB to
FD is given. But the reafon of the fame
FD to E is alfo given : Therefore the reafon
of HB to £ is in like manner given, and fa
is alfo the magnitude AE : Wherefore the mag-
1 1, def. nitude AB c is greater than E by a given mag-
nitude, and in reafon,

OTHER-

Digitized by Googl

EU GLIDE'* DATA. 379

t

OTHERWISE.

E ' F ConftruQion. Le£

- 1 : 1 ' B there be three mag-

C nitudes AB, C, and

D, and let AB be

0 greater than C by

' ' — ■ ■ a given magnitude,

and in reason ; but
Jet C be alfo greater than D, by a given magni-
tude, and in reafon : I fay that AB is greater
than D by a given magnitude, and in reafon.

Demonjtr. Forafmuch as AB is greater than,
C by a given magnitude, and in reafon, let the
given magnitude AE be cut off there-frqm :
Therefore the reafon of the remainder EB to
P is given. But the magnitude C is greater
than the magnitude D by a given magnitude^
and in reafon; therefore d EB is greater thandij. frof.
D by a given magnitude, and in reafon : Where-
fore let* the given magnitude EF be cut of%
there-from ; and the reafon of the remainder
tB to D fhall be given. But AF is e given.
Therefore /AB is greater than D by a given e i.prop.
magnitude, and in reafon. f n. ief.

PROP. XX.

5 G If there le two gi-

A" 1 - l B ven magnitudes, JB

J? and CD, and that

C 1 1 D from them there be

taken magnitudes JJS
and CF, having to one another a given reafon i
either the remaining magnitudes EB and FD, Jha%
have to one another given reafons j or elfe the one m-
Jball le greater than the other by a given mag-
nitudeand in reafon. s 4

Vtmonfc.

Digitized by Google

J8p

#

bip. 5-

EUCUDE'; DATA.

Demonjlr. For feeing that both the magni-
tudes AB and CD, are given, the reafon of the
faid AB to CD is a alfo given ; And if it be the
fame as of AE to CF, that of .the remainder
EB to the remainder FD lhall be b alfo the
fcme ; and therefore the reafon of the faid
EB to FD fliall be alfo given. But if it be
not the fame, let it be fo as that AE be to
CF, as AG to CD. Now the reafon of the
faid AE to CF is* given: Therefore the reafon
of the faid AG to CD is given. But CD is
given, therefore c AG is alio given. But the
whole AB is likewife given, therefore d the
remainder BG is given. And feeing that as
AE is to CF, fo is Afi to CD, and alfo the re-
mainder EG to the remainder FD, the reafon;
of the faid EG to FD is given. But GB is
alfo given : Therefore the magnitude EB is
. greater e than the magnitude BD by a given;
magnitude, and in realon.

c%.pop.

PROP. XXL

A

C

a
-l-

B
-i
D
-1

If there be two magnitudes
E given AB and CD, and to
them be added other magni-
F tudes BE and DF, having
to one anotJjer a given reafon:
Either the whole AE and CF JbaU have to one
m another a given reafon, or elfe the one Jball be
greater than the other by a given magnitude^ and
in reafon.

Demonjlr. For feeing that both the magnitudes
ZX.frof. ABandCD are given, their reafon a is alfo
given ; and if it be the fame reafon as of BE
to DF, the reafon of the whole AE to the
whole CF lhall be alfo given ; for it lhall
be b the fame. But if it be not the fame, let
it be as BE is to DF, fo BG to CD : There-
fore the reafon of the faid BG' to CD is given.

But

b iz. j.

Digitized by Google

EUCLIDE'x DATA ,8f

But CD is given, therefore c alfo BQL fliall be c z.pofi

given. But the whole AB is given, therefore

alfo the d remainder AG fliall be given. Andd A.propm

feeing that as BE is to DF, To is BG to CD,

and alfo e the whole GE to the whole CF,e 12. J*

the reafon of the faid GE to CF fhall be Hke-

wife given. But AG is given, therefore the

magnitude AE is greater than the magnitude

CF by a given magnitude, and in reafon.

«

PROP. XXII.

B If two magnitudes AB

* 1 - » C ana BC, have to Jbvie
D other magnitude D, a

— —- — - given reafon, alfo their

compound magnitude AC9
JhaU have to the fame magnitude D> a given
reafon.

Demonftr. For teeing that each magnitude AB
and BC, hath a given reafon to D, the reafon
a of AB to BC is given ; and by compounding, a g. prop,
h the reafon of AC to BC is given. But that b 8. W
of BC to D is alfo given, therefore c the rea- c 8 propi
fon of the faid AC to D fliall be likewife *r
given.

PROP. XXIII.

If the whole AB he to
the whole CD in a given
reafon, and that tbe parts
AE and EB, le to the
parts CF and FD in gi-
ven reafons, althol* they be not the fame, the whole
(to wit, AB, AE, and BE,) JhaU he to the whole
(to wit, CD, CF, and FX),) in given reafons.

Demonjtr. For feeing that AE is to CF in a
given reafon, let the fame be made of AB to
CG j therefore the reafon of the faid AB

to

382 EUCLIDE'* DATA.

a 19. J. to CG is given ; and confequently, alfo that a
of the reft EB to the reft FG. But the rea-
fon of FD to thefarfte EB is alfo given : There-

b %.frof. fore the reafon of FD to FG b is bkewife given j

c $.frop* and therefore c that of FD to the remainder
GD is alfo given. But the reafon of AB to
each of the magnitudes CD and CG is given :

AS. prof. Therefore d alfo the reafon of CD to CG is

e %.pop. g(iven, arid again e that of CD to the remain-
der GD. But the reafon of FD to DG isi

f 8. prof, given, therefore alfo /that of the fame CD to

FD, arid confequently

g $.prop. , E that of g CD to the re-

A ■ -1 — — B mainder FC \ and t here-
f G fore alfo the reafon of

C 1 — 1 D CF to FD fhall be gi-
ven. But the realori
Of EB to FD is propofed to , be given \ there-
fore the reafon of CF to EB fhall be giver*.
Again, for that the reafon of AB to CD is

fiven ; and alfo that of CD to each of thofe
C arid FD, the reafon of the Carrie AB to
hS.prop. each of the laid FC ai\d b FD, fhall be like-
wife given. But the reafon of the faid FD
to EB is given : Therefore the reafon of AB
to BE fhall be alfo given, arid corifequently
i $.frof. AB to the remainder i AE. Wherefore by
k/f J.5./ndivifion k the reafon of AE to EB fhall be like-
wife given. But the reafon of EB to FD is
given. Therefore alfo that of AE to FD. In
like manner, feeing that the reafon of CD to
AB is given £ arid that 6f AB to each of his
parts AE and EB ; alfo the reafon 6f the faid

1 8. Prop* t0 cac*1°,f AE and EB, / fhall be

given: Wherefore each of the magnitudes AB,
CD, AE, EB, CF, and FD, is to each of the
Others in a given reafon,

»

PROP.

Digitized by Google

EUCLIDE'* DATA* 585

PROP. XXIV.

■

1

A D If of three right lines

— ■ ■ A% B% and C, proportion

B F nal) J to By as E to C,

the firft A bath to tlft,
third C agiveri reafon\
it will atfo have to the
fecond £ a given reafon.
Demonftr. For, let there be expofed another
right line D, and feeing that the reafon of A to
C is given : Let the fame be made of D to F 5
therefore the reafon of D to F is given. But
D is given, therefore F is alfo given ; betwixt
the two right lines D and F, let there be taken
a a mean proportional E. Therefore the reft- - I? ^
angle made under D and F is equal b to the k -4 *
fquare of E. But the fame redlangle of D 7' a
and F is c given : (for all the angles of that c -
reftangle are given, being right angles, and the ^
reafons that the fides have to one another are
alfo given 5) therefore the fquare of E is given,
and confequeritly the fame right line E is alfo
given (for one equal thereto may be found, d d 14, 2*
feeing that the rettangle of D and F is given.)
But D is given, therefore e the reafon of D to e i.prop*
E is given, and as A is to C, fo D is to F.
But as A is to C, / fo the fquare of A is to f 1. &
the reftangle of A and C, arid alfo as D is to
F, fo the fquare of D is to the redlangle of
D and F. Therefore as the fquare of A is to
the reftangle of A and C, fo the fquare of D •
is to tl*e redtangle of D and F. But the redan-
gle of A and C is equal to the fquare of B,
(feeing that A, B, and C, are proportional)
and that of D and F to the fquare of E, there,
fore as the fquare of A is to the iquare of
B, fo the fquare of D is to the iquare of E :

Where-

Digitized by Google

5 84 EUCLIDE'* DATA.

; 12. 6. Wherefore g as A is to B, fo D is to E. But
1 z. def. the reafon of D to E is given, therefore b
alfo the reafon of A to B is given;

OTHERWISE. .
Vemonjlr. Forafmuch as the reafon of A to
C is given, and that as A is to C, fo the
fquare of A and C, the reafon of the faid fquare
ot A to the reftangle made of A and C, is alfo
given. But to that reftangle made of A and C
the fquare of B is equal (feeing that
A A, B, and C, are proportional;)
— therefore the reafon of the fquare
B of A to the fquare of B is given ^
■■ and by confequence, the reafon of

G the line A to the line B is given ;
, for to each of them A and B, we

have exhibited an equal to the

proper fquare ot each one.

» - 1 V'i

PROP. XXV.

If two lines AB and
CD, given by pofition
do interfed the point
E in which they inter-
fed: one another , is gi-
ven by pofition.
Demonftr. For if it change its place, the one
or the other of the lines AB and CD, would
change its pofition : But fo it is that by Sup-
a 4. def pofition it changethnot : Therefore a the point
E is given by pofition* ^

PROP. XXVI.

A — : B If the extremities A and B, of

a right line JB, U given by pofi-
tion* that fame right line AB is given by pofition
and by magnitude*

Vemonjtn

Digitized by Google

fetfClIDfi^ DATA. j8j

t)emonftr. For if the point A remaining in
its place, the pofition, or the magnitude of tbe
right line AB Ihall change, the point B win
fall elfewhete But fo it is, that by Supp0fu
tion it doth not fall el fe where. Therefore tile
right line AB is given by pofitbn* wnlb7
magnitude*

PROP. ■■iXXVTL

A B If one of the extremities J of

a right line AB^ given by fofitiort
tind magnitude be given, tlx other extrennty B
fiaH H alfo given.

Vemonfir. For if the point A remaining in
its place, the point B fhall change and fall in
Tome otheT place, either the pofition of the
right line AB, or its magnitude woiiid change :
But fo it is that according to the Suppofition, ■
neither the one nor the other doth change*
Therefore the point B is given.

OTHERWISE.

Conftr. On the center A *
•ttith thediftance AB, de-
fcribe the circumference
BCV r

Thmonfir. Thetsfore Az6.def
that circumference BC is
given by pdfitiori. But
the right line AB is alfo
given by pofition ; therefore the point * B isbi$*froj>*
given*

fe b . PROP*

Digitized b

EUCLIDEV DATA.
PROP. XXVIII.

tf hy a given point
A, there be drawn a
right line DAE, againfi
another right line BC9
given by pofition, the
right line DAE fo
drawn, is given by po-
rtion.

toeitionflr. For if it be not given, the point A
remaining in its place, the pofition of the right
line DAE may change : Let it then change it it
be pofiible, and fall elfewhere, remaining paral-
lel to BC, and let it be the line FAG : There-
fore BC is parallel to the faid line FAG. But
a the fame BC is alfo parallel to DAE :
Therefore b DAE is parallel to the faid line
FAG, which is abfurd j feeing they joyn to*
gethei, and meet in A : Therefore the pofition
of the Tight line DAE falls not elfewhere.
Wherefore the faid line DAE is given by pot -
tion. •

PROP, XXIX*

If to a right lint
JB, given by pofition1
and to a point C gi-
ven therein, there be
drawn a right line
CD, which Jball make
a given angle ACD, the line drawn CD, is given
by pofition.

Dcmonftr. For if it be not given by pofition,
the point C remaining in its place, the pofition
X)f the line CD obierving the magnitude of
the ancle ACD, will fall elfewhere. Let it
fall elfewhere then if it be pofiible, and let

m

Digitized by GooqI

EUCLIDFx DAT J. 387

itfce CE. Therefore the artgle ACD is equM
to the angle ACE, the greater to the leffer;
-which is abfurd. Therefore the pofition of the
right line CD, fhs*H not fall elfewhere * and
therefore the faid line QD is given. Dy po-
fition.

PROP. XXX. 1

If from a given joint AJ
be drawn to d right line BG?M
given by pofition, a right
line making a friven
aii^U Au&y that hnc drawn
AD is given by pofition.
. bemonfir. Fof if it Be
not given, the point A
remaining in its place,
the pofition of the right line AD keep-
ing , magnitude df the angle ADB , will
change. Let it change then, and let it be the
right line AE : Thetefore the angle ADB is
equal to the angle AEB, the greater a to the a i& tj
lefier, Which is abfurd. Therefore the pofition
of the right line AD doth not change ; and
therrfore the faid line AD is 'gfven by po-
fition. ^

OTHERWISE. \:

Confin By the point A let thefe b\ dr*w«
the line EAF, parallel to the right line BC.
Demonjkt Then teeing that by the given
oint A, and againft the right line BC, given:
y pofitiony there is dra^-n the l ight line bbj
thgfe lines EF and BC are parallels.- But o<i

B b % th«

Digitized by Google

3 88 • EUGLIDE'j DATA.

the fame lines doth
alfo fall' the right,
line AD, Therefore ,
b 20. U L» 1 Jj in j * the angle FAD is

equal to the given,
angle ADB ; ancL
therefore it is alfo given. Wherefore to the
right line Ht given by pofition, and to the
given point A therein, there is drawn the right
Efte AD, making the given angteFAD.
c i9*pop. Therefore c the faid line AD is given Dy po-
fition,

OTHERWISE.,

Conjtr* In the line BCE, let there be taken
the given point C, and by the fame let there
be drawn the line CF, parallel te the Xai&
DA.

Demonftu Forafmuch as AD and FGare^JK.
rallels, and that on them there doth ftUthe>
right line BCE, the angle FCB is equal d to <
d the given angle ADB 5 and therefore it is

alfo given. And feeing that the right; iine.BQ .

is given by pofition, and .
^ that to a given point C
therein, there is drawn .
the right line FC, ma-
king the given angle
_ _ FCB, that lame line FC

C ip. pOf. f t is giyen 5y pofltioa#

But Dy the given point
A, oppofite to the line
FC given by pofition, „

* « a t\ »»M ~ t^iere Is drawn the line

fa8. AD. ^Therefore the faid line /AD is given by .

OTHER*

Digitized by Google

EUCLIDF, DATA.

OTHERWISE.

1*9

Conjlr. In the right
line BC aiTume foine
point at F, and draw
AF.

Demovp.Yortfmuch
as ^ each point A and
F is given, the right
line AF is given g by
pofition. But theg 26.prof*
line BC is alfo. .gi-
ven by pofition. Therefore * the angle AFD is
given-. But by fuppofition, the angle ADF is
given : Therefore DAF (which is the refidue b 2% i
I of two right angles) is given ; and feeing * *
that to the right line AF given by pofition,
and to the given point therein A there is
drawn the right line DA, making the given iic.trtA
angle DAF, i that fame line £>A is given by
vpoiitigjj.

Scholium.

"* EtTCLIDE fuppofeth here that tm right
hnes being given by pofition, and inclining to one
another do make a given angle, which fome da de~
vionftrate after this manner*

Demonftr. Forafmuch as the two right lines
given by pofition, do incline to one another,
the inclination of thofe lines is given. But
the angle is the inclination of the lines: There-,
fore the angle which makes the right lines
given by polition, and incurring to one another,
is given.

Bb j

Jnc-

EUCLIDEY DATA.

Another thus jemonftramb it,

Conflr.Jjtt there be twp
right lilies inclining to
one another, as AB and
CB, given by poiitions
and in the line AB let
there be taken a given
point A, and in BCalfo
fome point, as and let
the right line AC $e
drawn.

Deptdnftr. Seeing that as
well the point B, *s each
of the points A and C is

taAuMi Siven> k the three right lines AB, BC, and AC,
K zo.projp. aj.e g£ven ^ magnitude. "Wherefore of three

direct lines equal unto them, a triangle toay
be conftituted : Let there then be made the
triangle FDE, haying the fide FD equal to the
fide AB, the lide FE equal to the fide AC,
and the bafe DE equal to tjie bafe BC.

Seeing then the angle? comgrifed of equal
right lines are equal, we have found the an-
gle FDE equal to the angle ABC 5 and there.
1 1. A/, fore thf fame / angle ABC is given.

PROP. XXXI.

If from & given point
A there be drawn tQ a
tight line given by poji-
tion BC9 a right line AD$
given hy yiagnituie, that
line AD JbaU hp a(fo gi-
ven by fofitioru
Conjtr. From the cen-
3jST^ — Tl ter A, with the diftance
AD, let the circle D££ be described.

Digitized by Google

r

b i$. prop

i6.prop*

• 2

EUCIIDE'j DATA. 391

Defnonjlr. Forafmuch as the center A is given
by pofition, and the femidiameter AD by mag.
nitude, the circle DEF a is given by pofition. a 6Jef.
But the right line BC Ms alfo give
fition : Therefore the point of interfe'
is given, and feeing tnat the point
given : c the right line AD is given
tion.

prop, xxxir.

Ifunto parallel right
Ivies JB and CD, gu
ven by pojitiony .there
be drawn a right line
EFy making the given
angles BEF and EFDn
■ ' the tine at awn r.r pan

G C be given by magnitude.
Conftr. For let there be taken in the line CD a x
given point G, and from that point let be.
drawn GH parallel to FE.

Demonjtr. Forafmuch as the lines EF and
HG are parallels, and that on them doth fall
the line CD 5 a the angle EFD is equal to the a 29. 1.
angle FGH. But the angle EFD is given,
therefore the angle FGH is alfo given* And
forafmuch as to the right line CD given by
pofition, and to the point G given in the fame,
there is drawn the right line GH, making the
given angle FGH, b the faid line GH is given b 19. prop
by pofition. But AB is alfo given bv pofi-
tion, therefore c the point H is given. But the c z^.prop.
point G is alfo given: Therefore d the lined z6. prof.
GH is given by magnitude, and is c equal to e $4. 1;
EF. Wherefore f tr*e faid tfn* EE is givenfi.^
by magnitude.

Bb 4

PROP,

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392

EUCLIDE'i DAT4.

\

XXXIII,

If unto parallel tight lines
J£ and CD, given by pofir
tion% there be drawn a right
line EF% given by magnitude,
that line EF JbalL make the
given angles BEF and DFE.

Conftr. For let there be
taken in the right line AB
the point G, $nd by that
point let there be drawn
rite line GH parallel to

, r Demonjlr. Therefore EF
a 54. i. is equal to the faid a GH. But EF is given
by magnitude, therefore GH is alfo given by
magnitude. But the point G is given, and
therefore if on that point, with the diftance
b6 def. GH, there be defcribed a circle, b that circle
ihall be given by pofition : Let it be then de-
scribed, and let it be HKL, the faid circle
HKL is therefore given by pofition. But the

vut • TTWl'lick ™th cut the circumference
AHL in H, is alfo given by pofition. Therefore
c is.frop.xhe laid point of interfeftion H c is given.
dz6.prop.Kut the point G is given: Therefore d the
right line GH is given by pofition. But the
right line CD is alio given by pofition : There,
e fch. 30. fore e the angle GHF is given. But to that
prop. angle / the angle EFD is equal 2 Therefore
f Z9. h the angle EFD is given ; and therefore alfo
the angle BEF 5 for that it is the refidue of
g 29.T. the lumm of twog right angles.

OTHERWISE?

Conftr. Let there be taken in the right line
CD the point G, and let GD be put equal to

EUCUDE'/ DATA. m

EF, then from the center G» with the diftance
GD, let there bedefcribed the circle DBH, and
draw GB.

nitude, the circle BDH b is given by pofitioa.h &def,
But the line AB is alfo given by pofitiont
Therefore i the point B is given. But the point i zj. frop ,

G is alfo given, there-
fore k the right line GB k z6.prop,
is given by polltion. But
the right line CD is aUb
givenbypofition: There-
tore /the angle BGD isl fcb. joi

fiven. Wherefore if EF^rof.
e parallel to BG, the
angle EFD m fhall be m zo. 1.
given, and confequently
alfo the other angle
BEF. But the right lines BG and EF being
not parallels, let them meet in the point H.
Forafmuch as EB is parallel to FG, and EF
is equal to GD, that is to fay, to BG ; alfp
FH n Ihall be equal to GH (for EH and BH p 14. 5,
being cut proportionally 0 by the parallel FG,o 2.6.
as EF is to FH, fo is BG to Gri ; and by
permutation, as EF is to BG, f o is FH to
GH :) Therefore f the angle HFG is equal to p $# %.
the angle HGF, but tte faid angle HGF i$
given (for that it is equal a to the given angle q 15. r.
BGD: ) Therefore the angle HFG is alfo gi-
ven. But to that angle the angle BEF is equal ;
and therefore is gi*en, as alio the remaining
angle EFQ. ^ " 1 y S

PROP,

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J94

EUCLIDE's DATA.

■

PROP. XXXIV.

r

Conftr. For
drawn the line EH, perp

If from a given point
E9 there be drawn unto
parallel right lines AB
and CD> given by pofi-
tion> a right line EFG,
that right line EFG Jball
be divided in agivenrea*
fonUowit) as EFtoFG.
the point E let there fc$,
ular to the lin*

Demonftr. Forafmuch as from the given point
E there is drawn to the line CD the right-
s' XQ.prop.line EH> making the given angle EHG a the,
laid line EH is given by pofition, but both
the oae and the other lines AB and CD is al-
b z$.prop. fo given by pofition. Therefore b the points
of interleftjon K and H, are given. But the
C z6. prop. point E is alfo given: Therefore c each line
d i.frqp. EK and KH is given. Wherefore d the rea-
fon of the faid fcK to KH is given. But as
. - EK is to KH, fo is EF to FG ; (for in the
triangle GEH the line KF being parallel to
HG, the fides EH and EG are cut proportio-
nally : ) Therefore the reafon of the faid EF
to FG is given.
.... . •

OTHERWISE. ,

Conftr. To the
K Parallel right
' lines given by
pofition,A B and
CD, let there be
drawn from the

M Cfc ' » po«» E the right
, r line FEG: I

lay that the reafon cf GE to EF is given.

Demanjlr^

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EUGLIDF* DATA. w

Bemonftr. For from the point E let there be *
drawn to CD the perpendicular EH, and pro-
duced to the point K; feeing therefore that from
the point E to the right line CD, given by
pofition, there is drawn the line EH, making
the given angle EHG, a the faid line EH is
given by pofition. But each line AB and CD a \o.prop.
is alfo given by pofition : Therefore b each
point of interfeftion H and K is given. But b zj.prop*
the point E is alfo given, therefore c each of
the lines EH and EK is given by magnitude ;e 26*. prop,
and therefore d the reafon of the faid EH to
EK is given. But e as EH is to EK, fo is EG d uprop.
to EF (for the oppofite angles at the point E e 4, 6.
being equal, and the lines AB and CD parallels,
the triangles EHG and EKF are equiangled;
and therefore as EH is to EG, fo is EK to EF ;
and by permutation as EH to EK, fo is EG to
EF.) Therefore the reafon of the faid lines
EG to EF is given.

/ PROP. XXXV.

If from a given point 4, to a right line BCf
given by pofition, there he drawn a right line JD9
which let he divided in E, in a given reafon (to
wit) of AE to EDy and that by the point of feftion
E9 there he drawn a right line FEG, ofpofite to the
right BC, given by pofition, the line FG drawn Jhall
be given by pofition.

Conftr. For from the point A, let there be
drawn the line AH, perpendicular to the line
BC.

Demonjlr. For feeing that from the given
point A there is drawn to BC given by pofition,
the right line AH making the given angle
AHD, a the faid line AH is given by pofition. a ^o.prop.
But BP is alfo given by pofition ; Therefore l b z^prop*
the point H is given. But the point A is alfo

given i

Google

59^ EUCLIDE'j DATA.

tz6.ptvp.ghtn : Therefore c the line AH is given by

magnitude and bypofi--
i«I tion. And feeing that

d as AE is to ED, fo is
AK to KH, and that the
leafon of AE to ED is
given, alfo the xeafori
of AK to KH is given >
and by compounding,^
theieafon of AH to AK
is given. But AH is
c C H D JBT gIve« by magnitude :

V2«F°f • * . a Therefore f alfo AK is

given by magnitude. But AK is alfo given
by pofition, and the point A is given : fheic
fi^.^fore^ the ; Point K is alfo given, and feeing

the line FG, oppofite to the right line BC
h z8.^.given by pofition 5 the faid line FG b is
given by pofition. *U ■

PRO P. X3TXVL

*

If from a given point J9
there he drawn to a nght Une
BC given by pofuion, a right
hne JD, and to it be added a
right line AE, having to tU
fame AD a given reaffn, and
3>f ™M b the extremity E of the

ff ad*ed line JE, there he drawn

a right line FEA, oppofite to
the unt BC, given by pofition,
that fame line FEK Jball he

r a r r glven h Potion.
Conjlr. For from the point A let there be
drawn to the line BC, the perpendicular AL,
and let it be prolonged to the point G.

«ofn?^rL Fo'afDluch as f">m the given
point A, there is drawn te the right line BC,

given

EUCLIDE'x DATA:

given by pofition, the right GL, which makes
the given angle GLD, a that line GL isa

fiven by pofition. But EC is alfo givcnb
y pofition, theretbie b the point L i? gi-
ven 5 and feeing that the point A is alfoc
given, the line c AL is given. But \\ rafmvch
as the reafon of AE to AD is given, and thatd
d as the faid AE is to AD, fo is AG to AL ;
(becaufe the triangles ALD and AGE are
equiangled) the realbn of AG to AL is alfo
given. But AL is given by magnitude : There- e
fore e AG is given by magnitude. But it is
alfo given by pofition, and the point A isf
given : Therefore / the point G is alfo given.
And feeing that by the fame given point G,
there is drawn the line FK, oppofite to theg
right line BC, given by pofition, g the faid
line FK is given by pofition.

PROP. XXXVII.

If unto parallel right
lines AE and CD, gi-
ven by pofition, there be
drawn a right Ine EF9
divided in the point G,
in a given reafon , (to
wit, of IG to GFj )
hut if by the point of
feftion G, there be
. . drawn oopofite to the

right lines AE or CD, given by pojtion, a right
line HGKy that line drawn Jhall be given by po-
sition* &

Confirm For let there be taken in the line AB
the given point L, and from that point let
there be drawn the line LN, perpendicular
to CD. r r

Demonfir. Seeing that from the given point
L, there is drawn to the right line CD, the

Wop
jo. prop;
z6.propm

2(5. prop*

4.(5.

1

9

2. prop*
27. propk1

i8.prop*

line^LN

1 making the given angle LND, the

faid

J by Google

j9g EUCLIDE'j D^r^,

a jo. prop, faid LN * given by pofition. But GD is
alfo given by pofition : Therefore the point

\i%$.prop. N h is given. But the point L is alfo given :

t z6. prop. Therefore c the line LN is given ; and feeing
that the reafon of FG to GE is given, and
that * as FG is to GE, fo is NM to ML*
the reafon of the faid MN to ML is given ;

d 6.prop* and in compounding, d the reafon of LN to
LM is alfo given. But LN is given by mag~

e 2, prop* nitude, therefore ML is e given by magnitude.
But it is alfo given by poution, and the point

f ijtprop. L is given : Therefore the point M / is alfo

Siven. And confidering that by the faid point
I there is drawn the right line KH, oppofite
to the right line CD* given by pofition, the
{aid line XH is alfo given by pofition*

Scholium.

*

* EUCLIDE fuppofeth here, that as FG is to
GE, fo NM is to ML s ha by another it is thus
iemonftrated.

The lines EF and LN are parallels §r not pa-
rallels : Let them in the firjt place he parallels,
and for of much as ly GonftruBion the lines EL9
FN, EF, and LN, are parallels, EN JbdK he *
parallelogram and therefore the fide- EF is equal
to the fide LN Jgain^ feeing that MG ,is paraU
lei to FN, ant, GF to MN, GN /ball he alfo a
parallelogram ; and therefore the fide GF is equal
to the fide MX. Wlterefore the equal fides EF
and LN, Jball pave to the equal fides FG arid
m + „ MN, g one and the fame reafon. Therefore as
* 7# * EF is to FG, fo is LN to MN ; and in dividing,

h 17. j* h *' c£ t0 GF> f° u LMt0

fro*

■

Digitized by Google

EUCLIDE'x DATA.

Now fupfofe that the lines EF and LN be not
parallels, but that they meet in the point 0. For*

afmucb

?9*

/rx iw the
triangle OFN there
is drawn HK, pa-
rallel to FN one of m
tin Jides $ i the fides i *• 6.
OF and ON are <fi-
videdpropottionably\
and tlmefore as FQ
is to CO, fo is NM
to MO. Again, fee*
ingthat in the tri-
angte OGM (here k
drawn EL, parallel to the fide GM, the fides
OG and OM arc divided proportionally : Wherefort
k as OE is to EG,fo is OL to LM^ and by com* kz.&
founding^ 1 as OG is to EG, fo is OMto hm\ hut
7t hath been demonftrated that as FG is to GO, fol i& $•
is NM to MO ; therefore in reafon of equality, m m %•
ms FG 7> to GE, fa it NM to MTt.

.PROP. XXXVIII.

. If unto parallel right
lines AB and CD, there
le drawn a right line
EF, and that to it there
he added fome other
right line EG, which
hath a given reafon to
the fame EF, but if by
the extremity G of the
added line EG, there! be drawn a right line HK%
againjl the parallels given by pojition AB and
CD, the line drawn HKjball be atjo given by fo-i
fition. '

Conftr. For let there be taken in tjie line
AB, the given point N, and from thence let

there

Digitized by Google

V

400

EUCLIDE'* r>^r^;

there be drawn to CD the perpendicular Nty*

-nd let it be prolonged to the point L.

Demonftr* Forafmuch as from the given point

N there is dtawn to the right line QD, given

by pofition, the right line NM making a

given angle NMF, the faid angle NMF a is

a jaj(wp.gjVen by pofition. But the line CD is alfa

, „ given by pofition : Therefore b the point M
bz$. pop.* iven' rBlit the point N is alfo giVen:

* 1^ trot Therefore c the line NM is given, and for
C20'Fithat the reafon of EG to EF is given, and
a r,h that d as EG is to EF, fo is LN to NM, the
tm Won of LN to NM is alfo given : But NM
e zprop *s givcn> therefore LN is e alio given. But
f vi prop *e point N is given : Thetefore /the point L
is alb given. Seeing then that by the given
point L there is drawn the tight line HK,

he line AB given by pofition, g
\K is alfo given by pofition*

faid

PROP. XXXDT-

- 1. ♦

If nl tlkfiiesofa triangle
ABC are given by magnitude,
the triangle is given by Kind.
Conftn For, let there be
)H cxpofed the right line DG
given by pofition, ending
in the point D ; but being
infinite towards the other
T part G> and therein let
x there be taken DE, equal
to AB.

Demonjtr. Now feeing the
faid AB is given by maj
tude, 1)E is fo alfo 5 but
the fame DE is alfo given
by pofition, and the point

a 2.7 trot) * PU1UAV"> *IIU lut rv

*7'*IX*'D is given : Therefore a the point E is given.

1 m

Again*

Digitized by Google

_ EUCLIDE'i DATA. r 401

Again, Let EF be put equal to BC ; and
feeing that BC is given by magnitude, EF lhall
be fo alfo. But the faid EF is in like manner y
given by pofition, and the point E is given :
Therefore b the point F is given. b 27.^00,

Furthermore, Let FG be taken equal tb
AC. Now forafmuch as the fajd AC is gi-
ven by magnitude, FG is fo alfo. But FG
is alfo given by pofition, and the point F is
given : Therefore the point G is alfo given*
Now from the center E, with the diitanct
ED, let there be defcribed the circle DHK, c c 6. def.
and that circle fhall be given by pofition*
Again, on the center F, and diftance FG, let
there be defcribed the circle GLK. There-
fore d the faid circle GLK Is given by pofi- d & fof
tioiij and therefore e the point of interfettion e z<. prop
K is given. But each of the points E and F
is given : Therefore each line/EK, EF, and f z6. prop
FK, is given by pofition and magnitude.
Therefore the triangle EKF is given * by
kind ; but it is equal and alike to the trian-
gle ABC ; and therefore the triangle ABC is
alfo given by kind.

Scholium.

* EUCLIDE fuppofeth
here that a tnavgie whof*
fides are given by magni-
tude and pofition, is gwen
by kind } but the antunt
Interpreters demonjhate it
in a manner thus. Foraf-
much as the right lines
KE and EF are given, g g t.frop*
the reafon which they have
to one another is given.
Alfo the right lines EF
and FK being given, their
reafon is alfo given \ and
in like manner , the reafon
C c cf

>

402 EUCLIDE'x DATA.

of the *J aid EKandFK is given. Again, feeing
that the fame lines KE and EF are given by
h fch. lo.pofition, h the angle KEF is given hy magnitude :
frop. Moreover , the right lines EF and FK being given
hy p option, the angle EFK is given hy magnitude,
as is alfo the rejidue EKF, and fo in the triangle
EKF are all the angles given, and alfo the rea-
i def. fons of the fides : Therefore i the faid triangle
EKF is given hy kind.

PROP. XL.

If the an-
gles of a
triangle A-
EC, are gi-
ven hy mag-
mtude, the
triangle' is

Jjl Jj given hy

Kind.

Ctwjln Let there be expofed the right line
DE, given by pofition and by magnitude ; and
let there be conftituted at the point D the
angle EDF, equal to the angle CBA \ but in
the point E the angle DEF, equal to the angle
BCA j therefore the third angle BAC is equal
to the third angle DFE,

Demonjlr. For each of the angles conftituted
in the points A, B, and C, is given : There-
fore each of thofe which are pofited in the
points D, F, and E, is alfo given $ and feeing
that to the right line DE given by pofition,
and to the point D given therein, there is
drawn the right line DF, which makes the gi-

a zg.prop. ven angle EDF, a the line DF is given by
pofition j and by the fame reaton, the line

bi^.p-of. EF is given by pofition : Therefore h the
point F is given by polition. But each of the

C i&profc points D and E is given : Therefore c each of

the

i

by Googlt

EUCLIDE'j DATA. 403

the lines DF, DE, and EF, is given by magni-
tude. Wherefore the triangle DFE is given
by kind ; and is alike to the triangle ABC :
Therefore the triangle ABC is given by kind.

- .

PROP. XLL

If a triangle ABC, hath one angle BAC given,
and that the two fides BA and AC, vthicb do
confiitute it, have to one another a given reafon,
the triangle is given by kind.

Conftr. For, let there be expofed -the right
Hoe DF given by magnitude and po\itior\. But
thereon, and at the given point F, let there
be conftituted the angle DFE equal to, the
angle BAC. -; •

Demonjlr. Now the angle BAC is given:
Therefore alfo the angle DFE is given, and ,
feeing that to the right line DF given by po- *
fition, and from the given point F therein, is
drawn a right line FE, making the given an-
gle DFE, a the laid line FE is given by poll-

tion. But feeing that the a
reafon of AB to AC is gi-
ven, let the fame be made
of DF to FE, then let DE
be arawn. Therefore the
reafon of DF to FE is .
given. But DF is given :
Thcrelore b FE is given by b 2. frop*
magnitude. But the lame *
FE is alfo given by pofi-
tion, and the point F is
given. Therefore c the c zj.frof*
point E is alfo given. But
each of the points D and
F is given : Therefore d each of the right lines d z6.pcf.
DF, FE, and DB, is given by poiition aid
magnitude, "Wherefore « the triangle DEF iae W
1/ . C c 1 give* 5*

4€>4 . feUGLIDE'i DATA.: N

given by kind. And feeing that the two tri-
angles ABC and DEF have an angle equal to an
angle, that is to fay, the angle BAC to the
angle DFE, and the fides which conftitute
f 6. d. thofe equal angles, proportional 5 f the trian-
gle ABC is alike to the triangle DEF. But
the triangle DEF is given by kind : There-
fore the triangle ABC is given by kind.

PROP. XLIL

If the fides of a triangle
y< JBCy be to one another in

given reafons, the triangle
F/r >3i ABC is given by kind.

« Conftr. For, Let there

be expofed the right line
D, given by magnitude*
and feeing that the rea-
fon of BG to AC is gi-
ven, let the fame be nude
ofD to E.

Demonftr. Now D is

a l.trov, 1 fM1 ! &Y™>. therefore a E is

* ■ - alfo given. Again, lee-

ing that the reafon of AC to AB is given, let
the fame be made of E to F. Now E is
hi.frop. given, therefore b F is alfo given. Now of
three right lines, equal to the three given
. . right lines D, E, and F, (and of which three
lines, two of them, in what manner foever
they be taken, are greater than the other :)
Let there be conftituted the triangle GHK, in
- fuch fort as D may be equal tp-JUK ; but E is
equal to KG, and GH equal 'to F; therefore
each of the. laid lines HK, KG, and GH, is
given by magnitude: Wherefore c tjie triangle
HGK is given by kind. A$d< feeing that as
• BC is to CA, fo is D to E^j apd that D i^
equal to HK, and.E.to>KG, as bC is to ( A,

fo

EUGLIDE'* DATA. 409

fo HK is to KG. Again, feeing that as CA
is to AB, fo is E to F, and that E is equal
to KG, and F to GH ; as CA is to AB, fo is
KG to GH. But it hath been demonstrated
that as BC is to CA, fo is HK to KG : There-
fore by reafon of equality, as BC is to AB,,
fo is HK to GH. Therefore d the triangle"
ABC is alfo given by kind.

PROP. XLIIL

5. 6. ,

If the fides BC and
BA^ about one of the tz-
aite angles of a reBau-
gled triangle ABC, have
to one another a given
reafon, that triangle is
given by kind,

Conjlr. Let there be
I ^ expofed the right line

*JE> DE given by magnitude

and pofnion, and on
X^N^ it let there be defcribed

Therefore a the femicir- a

cle DGE is given by pofnion.

Demonfir. For the line DE being given, and
divided in two equal parts, the center of the
faid circle is given by pofition, and the fe-
midiameter by magnitude. And forafmuch as
the reafon of BC to BA is given, let the
fame be made of DE to F : Therefore the rea-
fon of DE to F is given. But DE is given,
therefore F b is alfo given. Now BC is gieat* b
tr than c AB : Therefore ED is d alfo gieater c
than F. Let DG be fitted equal to F, and let d
EG be drawn | then on the center D, with the
diftance DG, let the circle GK be defcribed.
Now that circle e is given by pofnion, feeing e
that the center D is given, and the feniidia-

C c 1 meter

6. def.

1 .pop
19. I.
14. 5.

\»Vt •. •
6. def.

EUCLIDE'/ DATA.

"meter DG alio given by magnitude. But the
femicircle DGEis alfo given by pofition:
f z$.pof. Therefore / the point of interferon G is given.

But the poirus D and E are alfo given, therc-
g z6.proji.forQ g each of the right lines DE, DG, and
EG, is given by pofition and magnitude.
Wherefore /; the triangle DGE is given by
kind. And feeing that the triangles ABC and
DGE have an angle equal to an angle, to wit,
i J i- J. the right angle BAC to the right angle i DGE,
and the fides about the angles CBA and EDG
pr0portional. But each of the others ACB and
DEG are lefs than a right angle : Thofe trian-
k7« 6\ gles ABC and PEG k are alike. But the
triangle DGE i given by kind : Therefore
the triangle ABC is alfo given by kind.

PROP. XLIV.

If a triangle ABC> hath
one angle B gwen9 and
that the fides BA and AC,
about another angle BAC9
have to one another a gi-
ven reafon9 the triangU
ABC n given by kind.
6 E Conftr. Now the gi-

ven angle B is either acute or obtufe, (for it
was a right angle in the foregoing Propofi-
tion.) Let it be in the firft place acute, and
from the point A let AD be drawn perpendi-
cular to BC. » V • •
Dcmonftr. Therefore the angle ADB is gi-
ven : But the angle B is alfo given $ and there-
fore the third angle BAQ is given : Wherefore
40. prop, a the triangle ABD is given by kind ; and
g. lef. therefore b the reafon of BA to AD is given*
But the reafon of the fame BA to AC is alfo
c8.trop. given: Therefore c the reafon of AD to AC
is given, and the angle ADC is a right angle r

Where-

a

i by Google

EUGLIDE'/ DATA. 4*7

Wherefore the triangle i ACD is given by d ppcr
ki»d: Therefore e the angle C is given. Bute ydcj.

ISWSMS S angle' ABC be

and on the fide CB prolonged, let there be

drawn the perpendicular AD. pp .

Dcmonftr. Forafmuch as the angle Atf^ is
given, the angle ABD which follows it,. fhaU
Se given. But the angle ADB is alfo given :
Therefore the third angle DAB is given
Wherefore g the angle ABD is given by Tfcind 5 g 4*£*
* and thererore h the rea-"

1 fon of DA to AB is gi-
5^ ven. But the reafon or
AB to AC is alfo given : .
Therefore i the realon of *
DA to AC is given, and
the angle D is a right
angle : Therefore the tri-
angle DAC is given by
kind, and thererore the
angle ACB is given. But
the angle ABC is alio
given : Therefore the third angle BAC is gi-
ven. Wherefore the triangle ABC is given Dy
kind.

4*8

iUCLIDE'/ MTJ.

i \. -A Vi i(f -fit :

PROP. XtV.

H •

If a triangle ABC,
haw one angle BJC gi-
ven, and that the line
com founded of the two
fides JB and AC, about
the [aid given angle
BAC, hath to the otf
fide BC a given reaft
the triangle ABC '
ven by kind.
Conjtr. For, let the

Vgl-

b j.6\
c 18. $.

angle BAC be divided into two equal parts by
a7./y#« the line AD, therefore a the angle CAD is

given. -

Demonjir. Seeipg that as AB is to AC, fo
I is BD to CD ; by compounding^ c as the*^,;
line compounded of CAB is to CA, fo is BC
to CD, and by permutation, as the line com-
gaunded of CAB. is to CB, fo is£A to CD.
But the reafon of the line compounded of CAB
to BC is given \ therefore the reafoJi of "CA p
CD is alto given, and the angle CAD is gi-
d44.frop.ven- Therefore d the triangle ACD is given
^ 1 1 by kind, and therefore the angle C is given.
But the angle BAC is alfo given : Therefore
t40.prop.ihe third angle B is given: Wherefore e the
triangle ABC is given by kind.

given by
OTHERWISE.

Confir. Let BA be prolonged- direftly unto
the point D, in fuch fort as that AD may be
equal to AC, and let CD be joined.

• *

V &nov/!r.

Digj^zed by Google

EUCLIDEV DATA.

Dcmonjlr. Forafxnuch as the reafon of the
line compounded of CAB to CB is given, and
that AD is e^ual to AC, the reafon of the

whole line BD to BC is
- , given. But the angle

ADC is alio given, ior
it is the half of the gi-
ven angle BAC (foT that
the faid angle BAC/ is f
equal to the two inter-
nal angles ACD and
ADC, which are g e-g
qual to one another, be-h
ing the fides AC and
AD are equal :) Where-

i

ft

fore the triangle BDC h is given by kind? and
therefore the angle B is given. But the ang
BAC is alfo given : Therefore th$ remainii
angle ACB is given : Wherefore ? the triane
ABC is given by kind.

PROP. XLVI.

J 2. f.

J. 1.

| \

/

\

M

V

^t

If a triavgle ABC
hath one angle B given ,

fMs, AC and AB, about
another angle BAC, hath

to the other Ji 'Jc BC a
given reafon, the trian-
gle ABC is given by
kind.

Confir. For let the angle BAC be divided in-
to two equal parts, by the line AD.
* Demonftr. Therefore (as hath been fhewn in
the foregoing Propolition) the compound line
CAB is to CB, as AB is to BD. But the rea-
fon of the faid compound line CAB to CB is
given : Therefore alio the reafon of AB to BD

is

4io

EUCLIDE's DATA

is given. But the angle B is atfo given:
z 41. prop. Therefore the triangle AID a is given by kind 5
hi.def. an(* therefore b the angle BAD is given. But
the angle BAG is double to that of BAD ; and
therefore it is alfo given. Therefore the third
angle C is given. Wherefore the triangle ABC
is given by kind.

OTHERWISE.

.1

& J. def.

Conftr. Let BA be prolonged direftly, and
let AD be put equal to AC, and let CD be

2~:~J *

Demonjlr. Forafuiuch

as the reafon of the
line compounded of
CAB to CB is given,
and that AD is equal
to AC, the reafon of
BD to BC is given ;
ana the angle B is alfo
given: Therefore the
triangle CDB c is gi-
ven by kind ; and
therefore d the angle
D is given : Therefore the angle BAG which
is double to BDC, is alfo given : Wherefore
the other angle ACB is given ; and therefore
the triangle ABC is given by kind. \

PROP*

1 .

• 1

Google

EUCLIDE* DATA.

r

4ii

PROP. XLVII.

• • •

ReSiline figures or Aim,
CDf?, given by kind, an
divided into triangles given
by kind.

Conftr. For let the right
lines EB and EC be
drawn.

Demonjtr. Forafmuch
as the reftiline figure
ABCDE is given by
kind, the angle a BAE
is given, and the reafon of the fide AB to AE
is alfo given : Therefore b the triangle BAE
is given by kind. Wherefore the angle ABE
is given. But the whole angle ABC is alfo
given : Therefore c the remaining angle EBC
is given. But the reafon of the iide AB to
the fide BE, and alfo that of AB to BC is
given : Therefore d the reafon pf BC to BE is

!iven, and the angle CBE is alfo given :
herefore e the triangle BCE is given by © 4^»P°h
kind. By the fame^ilcourfe it may be demon-
ft rate d that , the triangle CDE is given by
kind. Therefore the reftiline figures given by
kind divide themfelves into triangles given
by kind.

PROP. XLVIII.

1 If on one and the fama
right line AB, are detai-
led triangles ae ACS and
ABD, given by f option,
tbofe triangles Jbali have
to one another a given
reafon, as ACS to ABD.

Confin

a S.J*
d &.prop*

Digitized by

4*2

b io.frop.
C 8. Prop.

e 8. /rop.
hi5. 5-

EUCLIDEV DATA.

Conflr. For from the points A and B, let
there be diawn at right angles on the line AB>
the lines AE and Bu, and prolonged unto the
points F and H j but by the points C and
P, let thete be drawn the lines ECG and
FDh | tU( I to AB.

DemJnjir, Forafmmch then as the triangle
ABC is given by kind, a the reafon of CA to
BA is given, and the angle CAB alfo given ;
but the angle BAE is given : Therefore the
remaining angle CAE is alro given ; but the
angle CAE is given j and tlie efore the other
angle ACE is alfo given. Whe efore b the
triangle AEC is given by kind. Now the rea-
fcn of EA to Ad c is given j (for d the rea-
fon of EA to AC, and that of AC to AB is
g^ven and in like manner, the realon of FA
to AB is given. Therefore e the reafon of
EA to AF is given ; but as AE is to AF, fo
/toe parallelogram AH to the parallelogram
AG ; but ACB is g the half of AH, and ADB
the half of AG ; therefore the reafon of the
triangle ACB to the triangle ADB is given,;
for it is the fame reafon with that of AH
to AG b j that L^p fay, g{ ^A to AF, whic^
is given.

PROP. XL1X.

c

i

If on one and the
fvne \\ght Une AB
there be described any
two reSibne figtires
JECFB and ADB ,
given by kind , they'
pall have to one ano-
ther a given reafon (to
wu) JECFB to ADB.

Conjtr. For let the
lines FA and FE be

draw^:

EUCLIDF* DATA:

drawn : Therefore each of the triangles a ABF,a
AFE, and ECF is given by kind.

Demonft). Seeing that on one and the fame
right line EF there are defcribed' the triangles
ECF and EAF, given by kind the-ieafon of
ECF to EAF b is given. Therefore by com-b
pounding, c the reafon of AECF to EAF is c
given. But the reafon of the faid EAF to
FAB is given, d being they are triangles d
given by kind, defcribed on one and the fame
right line AF : Therefore e the reafon of AE- e
CF to FAB is given. Wherefote by com-
pounding, /the reafon of AECFB to FAB isf
given. But the realon of the fame FAB to
ABD g is given : Therefore h the reafon ofg
AECFB to ABD is alfo given. h

47. prop.

4$ .prop.
6. prop.

48. prop*

8. prop.

6. prop*

JS.profe
» prop.

PROP. L.

i. *

i

If two fight lines
JB and CD 9 have to
one another a given
tea/on, and that on
thofe lines there be
defcribed reStiline fi-
gures AEB and CFD+
alike, and alike po-
fited9 they mil have to
one another a given
reafon.

Demonftr. To the
two lines AB and

CD, let there be taken a third proportional :
Therefore as AB is to CD, fo is CD to G.
But the reafon of AB to CD is given : There-
fore the reafon of CD to Q is alfo given :
Wherefore a the reafon of AB to G is given, a %.prop.
But b as AB is to G> fo is AEB to CFDrbcoMO,
Therefore the reafon of the fame AEB to CFD 20. 6.
is given,

PROP.

\

41*

• * • ' %

1

EUGLIDE'/ D^ZA

PROP. LI.

If two
right lints

A R mm* A

A B and
CD, have
to one ano-
ther a gi-
venreafon,
and thai
upon them
ppf there be
defcribed

any reftiline figures AEB and CFD, given by kind,
they will have to one another a jnven reafon% (to
wit, that of AEB to CFD.)

Conftr. For on AB let the reftangled figure
AH be defcribed alike and alike polited to
DF.

Demonjlr. Now DF is given by kind : There
fore alfo AH is given by kind- But AEB is
WO given by kind, and defcribed on the fame

a49.fr0p.tiae AB: Therefore a the reafon of AEB to
AH is given : And leeing that the reafon of
AB to CD is given, and that on thofe lines
are defcribed the re&iline figures AH and DF

b $o.prop. alike, and alike polited, the reafon b of the
faid line AH to DF is given. But the reafon
of AEB to AH is alfo given : Therefore the

c B.frof. reafon c of AEB to DF is given. T

PROP.

ri h

Google

EUCLIDE'x DATA*

PROP. L1I.

4T

If on a tight line JB$
given by magnitude, then
be defcnbed a figure aCB9
given by kind, that figure
JCB is given by magni-
tude*

Conjlr. For on the fame
line AB, let the fquare
AD be defcribed. There-
fore AD is given by
kind * and by magni-
tude.

Demonftr. Seeing that
on the right line AB, are defcribed the two ;
redliline figures ACB and AD, given by kind,
*" a the reafon of ACB to AD is given : There- a iQ.frof*
fore b ACB is given by magnitude. b z.frof.

Scholium.

4

* The antient Interpreter bath noted here thai
*very fquare is given hy kind ; for that ail the
angles thereof are given ; being all equal and right
angles : But alfo the reafons of the fides are given 5
for thofe fides being all equal, their reafons are
alfo equal. Moreover , whenfoever a fquare is ex-
fofed, a fquare equal thereto may be exhibited ;
and therefore the /quote is given by magnitude, a*
alfo each fide thereof

PROP.

i by Google

41*

EtfcXIDE'i DATA
PROP. Lfilf

a 3. def.
b 8. pro

c 3.<fe

J/ *Z>erf he two fi-
gures AD and EH, £i-
ven by kind, and that
one fide BD, of the one9
hath to a fide FHof the
other, a given reafon j
the other fides fbaU have
alfo to the other fides
given reajons.

Demonjlr. For feeling
that the reafon of BD
to FH is given, and
4lfo that a of BD to BA, h the reafon of the
laid AB to FH is given. But the reafon of the"
lame FH to FE c is alfo given : Therefore d
d 8. prof the reafon of AB to EF is given. In like man-
ner alfo the reafons of the other fides to the
other fides are given.

PROP. LIV.

If two figures A
and It, given hy kini,
have to one another a
given reafon, alfo
their fides Jball he to
one another in a gi-
ven reafon.

Conftr. For either
the figure A is
alike and alike po-
fited to B, or is not : Let it in the firft place
be alike, and alike pofittd , and let there be
taken the line G, a third proportional to the
lines CD and LF.

Vemonftn

Digitized by Google

EUCLtDE\c DATA. 417

Lemonftr. As CD is to G, a fo is A to B.a cor. tp,
But the reafon of A to B is given ; therefore xo. 6.
alfo the reafon of CD to G is given. And
feeing that CD, EF, and G, are proportional,
h alfo the reafon of CD to EF is given. Butb i^.pop,
A and B are given by kind : Therefore c thee jj, prop.
other fides fhall have given reafons to the 3
other fides.

Now let the figure A be not alike to the
figure B, and let there be defcribed on EF the
figure EH, alike and alike pofited to A :
Therefore the figure EH is given by kind ;
but the figure B is alfo given by kind : There-
fore d the reafon of B to EH is given ; and d 49. prop #
therefore the reafon of A to the fame EH e ise &pop.
alfo given : But A is alike to EH : Therefore
■ (by what is abovefaid) the reafon of CD to
: < EF is given ; and in like manner the reafon
of the other fides to the other fides is gften.

OTHERWISE.

Con/lr. Let there be
expofed the given line
GH: Now either the
figure A is alike to the
figure B, or not. Let
it in the firft place be
alike, and let it be as
CD is to EI?, fois GH
to LK ; then on GH
and LK let the figures
M and N be defcribed
alike, and alike pofited
to the faid A and B,

1*

K

N"

O

which figures M and N fhall be consequently
given by kind.

Demonftr. Therefore feeing that as CD is to
IF, fo is GH to LK, and that oh thofe lines
CD, EF, GH, and LK, are defcribed the figures

Dd A, B,

Digitized by Google

4i8 EUCLIDE'j DATA.

f 2.1. 6. A, B, M, and N, alike and alike pofited • / as
A is to B, fa is M to N. But the reafon of
' A to B is given : Therefore the reafon of M
g 52. tro/\to N is given, But g M is given, confidering
that 1 it is a figure givea by kind, defcribed
on a right line givea by magnitude j theretore
N is aJ?o given.

Confir. 2. Now, on LK let the fquare O be
h fch $z. defciibed-: Therefore /; the figure O is given

trof* by kind.

Demonflr. 2. Wherefore the reafon of O to
N is given. But N is given : Therefore O
ifch. 51. is given 5 and confequently, i alio KL. But
pop. GH is given : Therefore h the reafon of GH to
k' inpotu KL is given. But as GH is to LK, fo is CD
to EF. Therefore the reafon of CD to EF is
given ; and therefore the figures A and B being
1 53 p .op% given by kind, /the other fides of the fame
figures lhall alfo have to the other fides gi-
ven reafons. But if the figures be not alike,
the latter part of the demouftraiion here above
wuft be oblerved.

prop. fcr. —

to

A

On n j

D

If a Jfcace A be given
by kind, and by magni-
tude, the fides thereof
Jball be given by magni-
tude.

Confir. For, let the
right line BC given
by pofition and by
magnitude, be expo-
fed 5 and thereon let there be defcribed the
fpace D, alike and ahke pofued to A * there-
fore the laid fpace D is given by kind.

Bemovftr. For that it is defenbed on the
line BC, given by magnitude, it ib aifo a given;
** trat- by magnitude. But the figure A is alfo given :

There-

IB

Digitized by Googl

EUCUDE'i DATA. 419

Therefore I the reafon of A to D is giveiub j.^rcp.
But thofe figurts A and D are given by kind :
Therefore c me reafon ot the*iine EF to thee $4*f*of*
line BC is given; Btit BC is givea: Therefor*
d EF is alfo given. But the reafon of thed $4<fe/*
fame EF to F(f is given: Therefore eFG ise %.po^
given. And by the fame reafons it in*y be
demonftrated that each of the other fides are
given by magnitude. ■

OTHERWISE,

Conjtr. Let th*
fpace GHIKL b*
given by kind And
by magnitude : I
fay that the fides
thereof are given
by magnitude, tot •
on die light linfc
GH let there be
defcribed the fquare
GM 5 therefore /fyfcj. jt»
GM is given by^.
kindt

Dcvionftr. But the
tyace GHIKL is
alfo given by kind :
therefore g the ieafon of the lame fpace GK g ^c*pop
to GM is given. But GK is given by magni-
tude : Therefore b GM is alfo givtn by magni- h %,frpp.
tude 5 and teeing thfct GM is thefquaie of the
line GH, i that line GH is given by magni \ pflh jj4
tudev Wherefore in like manner, each of the prof,
other lines HI, IK, KL, and LG, is given.

D d %

P R OP.

Digitized by Google

4^0

EUCLIDE'i DATA.

a J4- 1*

••0

b 14. 6,

CT. 6.

PROP. LVL

If two eqinangled parallel
logrami A and B, have to
one another a given reafon*
as one fide CD of the firjt Jt
is to one fide FG> of the Se-
cond B \ fo the other fide
GE* of the fecond B* is to
that to which DH the other
fide of the firft A* hath the
given reafon that the parallelogram J hath to the
■oarallelogyam B%

Confir. For let HD be prolonged direftly to L,
fo that as CD is to FG, lo HD may be to
DL } and finifh the parallelogram DK.

Demonjlr. Seeing that as CD is to FG, fo
HD is to DL, and a that CD is equal to KL ;
as LK is to FG, fo is GE to DL ; and thus
the fides about the equal angles DLK and
EGF are reciprocally proportional : Wherefore
I DK is equal to B ; and therefore feeing the
reafon of A to B is given, and that B is equal
to DK, the reafon or A to DK is given. But
as c A is to DK (that is to B) fo is HD to
DL ; therefore the reafon of HD to DL is
alfo given : and feeing that as CD is to FG,
fo GE is to DL, and that the right line HD
hath to DL a given reafon ; to wit, that
which the fpace A hath to the fpace B ^ as
CD is to FG, fo GE is to that to which HD
hath the given reafon that the fpace A hath
to the fpace B, that is to fay, the reafon of
HD to DL.

PROP.

Digitized by Google

EUCLIDE'x DATA. 421

PROP- LVII.

If a given Jfcace JD be applied to a given right
line JB in a given .angle CAB, the breadth CA
of the application is given.

Conftr. For on AB, let there be defcribed
the fquare AF \ therefore a the fame AFisayf#.
given : Let the lines EA, FB, and CD, beprop.
prolonged to the points G and H.

Demonjlr. Seeing therefore that each fpace AD
and AF is given, their reafon is alfo given.
But b AD is equal to AH: Therefore the rea-b j6.r.
fon of AF to AH is given : Wherefore the
jreafon of EA to AG is given, (for c it is thec u&

fame with that of AF to
AH.) But EA is equal to
AB ; theiefore the reafon §
of AB to AG is given.
Now Teeing t^hat the angle
CAB is given, and the an-
gle GAB alfo given, the
refidue CAG is given. But
the angle CGA is alfo gi-
ven, being a right angle; Therefore the re-
maining angle ACG is given. Wherefore the
triangle^ CAG is given by kind. Therefore d 4c. prop.
the reafon of CA to AG is given. Eut the
reafon of AB to the fame AG is alio given :
Therefore the reafon of CA to AB is given ;
and the faid AB is given : Wherefore CA is
alfo given.

l>4 J

PROP.

«

Digitized by

4**

EUCLIDE'* DATA.

PROP. LVIII.

If a gi-
ven [face

AB, be ap-
plied to a
given
right lint

AC, want-
ing by a fi-
gure DEy
given by

kind, the breadths of the iefecls 6te given.

Covflr. For let AC be divided in two equal
parts in the point F : Therefore as well AF
as FC is given. On the faid line FC let there
be dele ri bed the reftangled figure FG alike and
alike pofited to DE. Therefore FG is given
"by kind. . * «

Demonjlr. Seeing the figure FG is defcribed
on the right line FC given by magnitude, the
\zjTj0£'isL]& rectiline FG is a alfo given by magnitude.
But FG is equal to AB and IL ; (for b AI
and FE being equal, and c FB and BG alfo
equal, the Gnomon ICL is equal to AB ; and
** therefore their added figure IL, common to

both, FG fhall be equal to AB and IL :)
Therefore the figures AB and IL together are
given by magnitude. But AB is given by
A 4. p> op. magnitude : Therefore d the remaining figure
IL is alfo given by magnitude. But it is alfo
e 2,4. 6. given by kind, feeing it is e alike to DE:
i 5 $*'prop. Therefore / the fides of the fame IL are
given : Wherefore IB is given ; and feeing
' g 54* *• tnat *r ls equal g to FD, the fame FD is al-
( i\q.prop. fo given. But FC is given, therefore the re-
i 5. ief* mainder DC /; is given 5 and i in a given
k 1.. prop, reafon to BO, and therefore k BD is given.

PROP,

b j6. 1
c4j. 1.

Digitized by

EUCLIDE'i DATA,

4*3

PROP. LIX.

Vi

»rc nr., ^ „

ven Jfiacc
JB he Ap-
plied ae~
cording to

1-1 y ' it> a green

K " right l.ne
I ex-*

ceeding it
hy a fi-
gure CB given hy kind, the h eadths of the excejjet
CEand CF are given.

Confir. For D£ being divided into two
equal parts in G, let there be defcribed on
GE the reftiline figure GH, alike and alike
pofited to CB.

Demonjtr. Now feeing that CB is alike to
GH, thofe figures CB and GH * are about
one and the lame diameter, and GH is given
by kind, as is CB. But it is defcrijjed on the
given line GE : Therefore a the fame GH is a %i*frofc
alfo given by magnitude. But AB is given :
Therefore AB and GH are given by magni-
tude. Now thdfe figures AH and GH, are
equal to LI, (for AGp LE, and EI, being e-
qual, the Gnomon GFH is equal tp AB; and
therefore adding GH common to both, LI
fhall be equal to AB and GH :) therefore LI
is given by magnitude j but lit is alfo given
by Kind, being ic is b alike to CB. Therefore b *4#^
c the fides of the faid LI are given, feeing P 5$ prop.
it is equal to GE : Therefore d the remainder d 4*prop.
CF is given, and in a given reafou e to Cfc. e 3 fi*U
Wherefore / Cii is given, 4 pop.

D d 4

Scrip-

-

Digitized by Google

EUCLlDE'i J) ATA.

Scholium.

< EUCLIDE fup-
pofeth here that CB
and GH are about one
.''and the fame diameter ^
but we JbaU thus de-
rnonflrate it : LetCB
and GH be two ai.ke
V parallelograms dijbofyd

\j " — "IB above , that is to
y fay, * that the equal
angles join together in
2?, the fide CE meets direffly with his homologal
fide EH, and the fide ME, his correjfondent fide
EG; and let the diameter FE be drawn, I fam
that the faid diameter FE prolonged, wiU pafs fy
the point K ; that is to fay, the varaUelograms GH
and CB, confift about one and the fame diameter.
For if it be denied, the diameter EF being pro~
faced, will pafs above the point K, or below it.
Let it in the firjl place pafs above it, and let it cut
GK, prolonged m the point M, and by the point M,
let there be drawn MN, parallel to KH, which
JbaU meet EH, prolonged in the point N, and FB
in 0. {{'; • fjfj

Demonjlr. Forafmuch as the parallelograms
GN and CB are with the parallelogram LO
about one and the /apie diameter, they are g
alike to one another, Wherefore as FC is to
CE, To is EG to GM. In like planner, feeing
\\k parallelograms CB and GH are alike, as
* 0 is to CE, fp is fcG to GJC : Therefore b
as EG is to'GM, l*o 4s EG to GK. Where-
foie i GM and QK are equal, a part to the
whole, which is abfyui : By tjie fame rea-
loiis it may be demonstrated ; that the diame-
ter prolonged will not fail below the point K :

■ n There?

Digitized by GoogI

EUCLIDFx DATA.

Therefore the parallelograms CB and GE con-
lift about one and the fame diameter.

4*i>

PROP- IX

CE and DG) Art given.

If a tarallelo-

gram ABy given
by kind and by
magnitude , be
augmen'ed or di-
vi inijbed by a Gno»
vion LtD y the
breadths of 'the
Gnomon (conjtjt.
ing of the lines

Demonjlr. For feeing that AB is given, and
the Gnomon CFD ajub given, the whole pa-
rallelogram BF is given : But it is alfo given
by kind, feeing it is alike to BA : Therefore
a the fides of the fam^ BF are given; and a 5s
therefore each of the lines BE and BG is
given. But each of the lines BC and BD is
given ; therefore each of the remaining lines
CE and DG is alfq given.

Conjlr. Now

*r\ let the paralle-

J logramBF, gi-

ven by kind and
■ by magnitude,

^A. \rt be diminiflied

by the given
Gnomon CFD :
I fay that each
J) J± of the lines CE

and DG is gi-
ven.

Vemonjlr. 2. For feeing that BF is given,
and the Gnomon CFD given, the remaining
jiguie i\B is alfo given. But it js pllO given

by

•frop.

4 1

Digitized by Google

tUCLIDE's DATA.

by kind, feeing it is alike to BF : Therefore
b tf.pop.b the fides pf the faid AB are given, and there-
fore each of the lines CB and BD is given.
But each of the lines BE and BG is given :
Therefore alfo each pf the remaining lines C£
and DG is given, f *

i •

PROP. LXI. \

L

If to one fide of a fi-
gure ABCE, given by kind,
there be applied a fpace pa-
rallehgram CD, in a given
angle ECFy and that the
given figure AC hath to tlx
parallelogiam CD a given
reafon, the parallelogram
CD is given by kind.

Conjtr. For by the point
B, let BH be drawn pa-
ly " ■ t — ^ rallel to CE, and by the
Js JV. Ji ft p0int E let EH be
drawn parallel to CB, and let EC and HB be
prolonged to the points K and G.

Demonflr. Forafmuch as the angle BCE is
Z \Mf. given, and the reafon of EC to CB, a the pa-
rallelogram CH is given * by kind. But the
figure ABCE is alio given by kind, and is
defcribed on the fame line BC, as the paral-
lelogram CH given by kind is : Therefore b
b 49-froP*he reafon of the figure ABCE to the pa-
rallelogram CH is given. But by fuppofitioi,
the reafon of the faid figure ABCE to the pa
C %6. i. rallelogram CD is alfo given; and CD is c
d 8. prop. cquai t0 CG : Therefore d the reafon of CH
to CG is given. Wherefore the reafon of
c 1.6. the line EC to the line CK is given \ (for e
as CH is to CG, fo is EC to CK.) But the
reafoTi of EC to CB is alfo given : Therefore
f 8. prop, /the reafon of the faid CB to CK is given.

And

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EUCLIDE'i DATA.

4*7

And feeing that the angle EC? is given/ alfo
the following angle BCK g is given. But theg ij.t.
angle BCF is propoled given ^ and therefore ^.prop.
the remaining angle FCK is given. Alfo the
angle CKF is given, for that h it is equal h 29. r.
to the angle BCK : Therefore the other angle
CFK is given ; Wherefore i the triangle FCK i 40. prop.
is given by kind 5 and therefore the reafon
of FC to CK is given. But the reafon of CB
to the fame CK is alfo given .* Therefore k k 8. prop.
the reafon of FC to CB is given 5 and the
angle BCF is alfo given. Wherefore the pa-
^"-•--Ttni CD is given by kirji

. . Scholium.

* * Altho* it le manifefl that a parallelogram that
hath one angle given, and the reafon of the fidcf
about the fame angle alfo giveny is given hy
kind,' as Euclide doth here declare , fo it is not-
withjtandmg that the antieitt Interpreter doth this
iemonftrate it.

Seeing that in the parallelogram CH tlx angle
ECB is given, the angle CEH is alfo given ; for
the right line EC falling on the parallels EH and
CB, doth make the two internal angles on the
fame part equAl to two right angles. And there-
fore feeing that the angle ECB is given, /£*
other angles are given ; and feeing that the reafon

°* t0 CB is &v™> and tliat EH « «f «4
°W another u alfo f^ven.

< » •

1 • ♦

*

prop.

Digitized by

4*8

EUCUDE'f DATA.

PROP. LXH.

If two

right lines
M and
CD, have
to one
another a
givenrea-
fon9 and
that on
one of
them JB%

, j there be

defmhei a figure AEB, given ly kind ^ but on
the other CD, a [pace parallelogram DF, in a
given angle DCF, and that ihe figure AEB hath

tauetogram DF is given by kind,

Conjtr. For on the line AB let there be de-
ferred the parallelogram AH, alike and alike
pofited to DF.

Demonftr. Seeing then that the reafon of AB

j°r •? jS ?iven> and that on thofc lines are
derenbed the reftiline figures AH and FD,

a $o.prop. alike and alike pofited, a the reafon of AH
to FD is given. But the reafon of FD to AEB

b 8. prop, is alfo given : Therefore h the reafon of AH
to AEB is given. But the angle ABH is ak
lo given, being eoual to the angle FCD, and
fo the figure AEB is given by kind ; and to
AB one of the fides thereof, the parallelogram
AH is applied in a given angle ABH, and

c6r.^.the^afon of the Caid figure AEB to the faid
r r para elogram AH is given: Therefore c the
paialleiogram AH is given by kind; and
thereto^ FD which is alike, thereto, is alfo
gi ven by kind.

PROP,

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EUCUDFi D4TA.

4*9

PROP. LXIII.

•

If & triangle
ABC be given
hy kind , the
fquare BE, CD,
and CF, which
is defaibed on
each of the fides,
JbalL have a gi-
ven teafon to
the mangle
ABC.

BeTnonJh\YoM
feeing that on
one and the
fame right line
EC, there are

defcribed the two reftiline figures ABC and
CD, given by kind, a the reaTon of the Came a
ABC to CD is given ; and therefore the
reafon of the fquares BE and CF, to the
triangle ABC, is alfo given.

PROP. LXIV.

If a Uiangle ABC, hath
an obtufe angle ABC gi-
ven, that Jpace of which
the fide AC fubt ending the
obtufe angle ABCy is more
in power than the fides JB
and BC, that comprehend
the faid angle, JhaU have
a given reafon to the tri-

ii -Ll Conftr. Let the line
CB be prolonged dire&ly, and from the point
A let the perpendicular AD be drawn: I

fay

Digitized

45o EUCLIDE'j DATA.

fay that the fpace of which the fquare of the
line AC doth eSceed the fquares of the lines

a 12, 2. AB and BC, that is to fay, a the double of
the reftangle contained under CB and BDj
fhall have a given reafon to the triangle ABC.

Demonjlr. For feeing that the angle ABC is
giren, the angle ABD is alfo given. But the
angle ADB is alfo given therefore the other

b 4a prop, angle BAD is given : Wherefore b the triangle

C7 def ABD is given by kind therefore c the realon
5' of AD to DB is given. Biit as, AD to DB, fo

d 1. 6. d the re&angle of AD and BC h to the yeftan-
gle of BC and BD. But the reafon of AD to
BD is given : Therefore- alfo is the reafon of
the redangle of AD and BC to the reftangle
of BC and BD given : Wherefore the realon
of the double of the faid rectangle BC and
BD to the reftangle of AD and BC is alfo
given. But the faid re&angle of AD and
BC hath alfo a given reafon to the triangle*
ABC (to wit, double reafon $ for the rediangle

e 41. i. is e double to the triangle) therefore the rea.

fon ofcthe. double of the reftangle of BC and

f&pop BD/to the triangle ABC is given. But the
fame double of the reftangle of CB and BD
is tha.t fpace of which the fquare of the
line AC doth exceed the fquares of the lines
AB and BC : Therefore the fame fpace hath
a given reafon to the triangle ABC.

» ■

■

«..«... »
mm •»•...»

• » •

«

4 • •> • • * ■

• » - ..»..• • •

«. -* % ' • » • • • •A

\

Digitized by Google

EUCLIDE'/ DATA.

4?i

.1 )

B D

prop, lxv:

// d triangle AKC, hath
one acute angle .JCB given,
that $acey of which the fide
fub tending the faid acute
angle is lefs An fewer than
the [ides comprehending the
fame acute angle, Jhall have
a given reafbn to tlx tri-
angle.

Conftr. From the point
A let there be drawn the
line AD, perpendicular to BC : I lay thatfpace
of which the fquare of the line AB is lefs
than the fquares of the lines AC and CB,
that is to fay, a the double of the rectangle a 15.2.
of BC and CD, hath a given realon to the
triangle ABC.

Bemonflr. For feeing that the angle C is
given, and the angle ADC alio given, the
other angle DAC is given : Wherefore the
triangle h ADC is given by kind ; and tru*re-fo40 pro*
fore the realon of AD to DC is given, and 4 P*
confequently alio c that of the rectangle of *
BC and CD to the rectangle of BC and AD : ,0*
Therefore the reafon of the double of the rect-
angle of BC and CD to the rectangle or BC
and AD is given. But the reafon of the fame
reftangle of BC and AD to the triangle ABC
is given (for d the redangle is double to the
triangle : ) Therefore e the reafon of the Uou- d 41.1-
ble of the redtangle of BC and CD to the** 8.frcp.
triangle ABC is given. And feeing that the
fame double of the rectangle of BC and CD
is tharvfjhereof the fquare of the line AE is
lefs than the fquares of the lines AC and BCf
that fpace of which the fauare of the line

AB

'I

tized by Google

4 J* EUCLIDE'i DATA

ABislefs than the fquares of the lines AC
and BC, fliall have a given reafon to the tri «
angle ABC. t

•PROP- LXVI.

If a 'triangle ACBS
hath one angle B given,
the reBangle made of the
lines JB and BC, con-
taining the fame angle^
fhall have a given reafon
to the triangle. *"
t Conftr. For from the
point A let AD be
drawn perpendicular to
CB.

Detnonjlr. Therefore feeing that the angle B
is given, and alfo the angle ADB ; the other
angle BAD is likewife given. Wherefore the
a 4a prof. triangle ADB a is given by kind; and confe-
miently the reafon of AB to AD is given.
But as AB is to AD, h fo the Wangle of
AC and CB is to the reftangle of CB and
AD ; Therefore the reafon of the Wangle of
■? AC and CB to the reftangle of CB and AD
is given. But the reafon of the faid reftan-
gle of CB and AD to the triangle ACB is

trnif air° givcn 5 (for that iz is *>uble rcafon, the
a q111+ redangle being double c to the triangle : )
*-FVm Therefore d the reafon of the reftangie *? AC
and CB to the triangle ABD is given*

b 1.6.

PROP

9 '

Digitized by Google

EUCLIDE'j data

PROP. LXVII.

If a triangle ABC
hath one aqgle B AC gi-
ven, that jfiace by which
the fquare of the line
compounded of the two
fides BA and A$, that
contain the fame given an-
gle BAC doth exceed the
fquare of the other fid*>
itjball have a given rea*
fon to the triangle ABC,
Conftr. For let BA be prolonged in iuch fort
is that AD may be equal to AC* then ha-
ving drawn DCE infinitely, from the point B
let BE be drawn parallel to AC, meeting
the faid DE in the point E.

Jfremonftr. Fofafmuch as AD is equal to AC,
d DBis equal to BE; (for the two triangles a 4*
ADC and BDE are alike) and from the top B %Z 5
is drawn to the bafe DE, the right line BC .»
Therefore * the reftangie of DC and CE, with
the fquare of BC, is equal to the fquare of
BD 5 but the fame BD is compounded of BA
and AC ; therefore the fquare of the compoun4
of AB and AC is greater than the fquare of
BC, of the rectangle of DC and CE.

Now I fay that the redangle of DC and CE
hath a given reafon to the triangle ABC :
Forafmuch as the angle BAG is given, the
angle DAC is alfo given. But each of the
angles ADC and ACD is given, it being the
half of the angle BAC which is given. There-
fore h the triangle ADC is given by kind ;l>4o
and therefore the reafon of DA to DC is
given* Therefore c the reafon of the fquare^
of the M DA to the fquare of DC iff alfoC5°

E e given*

4H

a i. 6.

ei. 6.

Si. 6.

EUCLIDE'i DATA,

♦

given. And feeing that as BA is to AD, i

fo is EC to CD, and
alfo as BA is to AD,
e fo is the redangle of
BA and AD to the
fquare of AD $ and as
EC is to CD, /fo alfo
is the redangle of EC
and CD to the fqiiare
of CD $ by permutation,
as the redangle of BA
and AD is to the red-
angle ©f EC and CD,
fo is the fquare of AD to the fquare of DC.
But the reafon of the faid fquare of AD ta
the fquare of DC is given : Therefore the rea-
fon of the redangle of BA and AD to the
redangle of EG and CD is alfo given. But
At) is equal to AC Therefore the reafon of
the redangle of BA and AC to the redangle
, - . of EC and CD is given. But the reafon of
the redangle of BA and AC to the triangle
g (6. prop. ABC g is given, becaufe the angle BAC is
ii 8. prop* given : Therefore h the reafon of the redangle
EC and CD to the triangle ABC is given*
But the redangle of EC and CD is that
•whereof the fquare of the line compounded of
BA and AC is greater than the fquare of BC :
Therefore that ipace to which the fquare of
the line compounded of BA and AC is greater
than the fquare of BC, fhall have a given
reafon to the triangle ABC.

» ■

Scholium. < -

* EUCLID E fvppofeth in this place that when
hi an Jfofceles triangle a - right line is drawn
from the top to the hafe^ the fquare of that line,
with the rcftangle contained under the fegments of

the

Digitized by Google

EUCLIDE', DATA. ^

He bafes, is equal to the fquare of either of
the other legsy which the antient interpreter doth
thus demonjtrate.

Conftr. Let ABC be an Ifofceles triangle,
whofe legs are AB and AC ; and from the top
A let AD be drawn to the bafe BC : I fay-
that the fquare of AD with the reftangle of
BD and DC, is equal to the fquare of either
Of the legs AB or AC.
DemonjYr. Now the line AD is perpendicu-
lar to BD, or not : Let
it in the firft place be
perpendicular : Therefore
it will cut the bafe BC
into two equal parts in
the point D ; and there*
fore the re&angle con-
tained under BD and DC
is equal to the fquare of
the laid BD, and adding
to them the common
fquare of AD, the reft-
angle of BD a^d DC with the fquare of AD,
fhail be equal to the fquares of DB and AD:
But to thofe fquares ol AD and DB i the i 47. r-
fquare of AB is equal : Therefore the fquare
of AB is equal to the re&angle cf BD and
DC, and the fquare of AD together.

Now fuppofe AD not to be perpendicular,
but that from the point A there ooth fall on
BC the perpendicular AE, that being To, BC
ihall be cue into Wo parts equally in the"
point E, and uneuually in D. Wherefore the
re&angle of BD and DC, with the fquare of
DE, k is equal to the fquare of BE j and k 2.*
adding the common fquare of AE, the rectan-
gle 6f BD and DC, with the fquares of DE
and AE, fhall be equal to the fquares of BE
and AE< But / the fquare of AD is equal to 1 47. u
the two fquares of DE and AE 2 Therefore
* , E e l the

Digitized by Google

4;6 'EUCLIDE'j DATA.

the re&angle of BD and DC, with the fquate
of AD is equal to the fquares of BE and
AE. But to thefe fauares of BE and AE the
fquare of AB is equal : Therefore the fquare
of AD, with the reftangle of BD and DC,
is equal to the fquare of AB.

OTHERWISE.

Qmftr* Having done, as in the foregoing
Demonftration, from the point A, let AF be
drawn perpendicular to CD, and let AE be
drawn.

Demonftr. Forafmuch as the angle BAC is
given, the half thereof ACF lhall be alfo
given. But the angle AFC is given : and
therefore the triangle AFC is given by kind:
Therefore the leaion of AF to FC is given.
But the reafon of CD to the fame FC is
alfo given, feeing that CD is double to FC :
m 8. Prop. Therefore m the reafon of CD to AF is given \
and therefore alfo the reafon of the redangle
of CD and EC, to the reftangle «f AF and
n i. <5. EC, is given ; (for it is the fame reafon n as

that of CD to hS\
But the reafon of
the redangleofAF
and FC to the xxu
angle ACE is gi-
ven ; feeing it is

041.1* k\S double 0 to the

fame triangle.
Therefore the rea-
fon of the redan*
gleof CD and CE
to the triangle
^ ACE is alfo given.
But the triangle ACE is equal to the triangle
p 27,1. ABC J), they being both conftituted on one
and the fame bafe AC, and between the fame

parallels

Digitized by Google

EUCLIDE', DATA. 437

parallels AC and BE : Therefore q the reafonq 8.fr*p.
of the reftangle of CE and CD to the trian-
gle ABC is given. But the faid reftangle of
CE and CD is the fpace by which the fquare
of the line compounded of AB and AC, is
greater than the lauare of BC : Therefore that
tpace by which tnfe fquare of the line com-
pounded of AB and AC is greater than the
fquare of BCf hath a given leafon to the tri-
angle ABC.

OTHERWISE,

For the given
angle A is ei-
ther aright, a-
cute, or obtufe
angle: Let it
in the firft place
be fuppoted a
right angle :
Therefore the
fquare of the
line compound-
ed of BAC, is
greater than the fquare of BC, by twice the
reftangle of BA and AC 5 (feeing that r the r 47. r.
fquare of BC is equal to the fquares of BA
and AC ; and the fquare of the line com-
pounded of BAC s is equal to thofe two s J. 2.
fquares of B A and AC, and twice the reftan-
gle of the faid BA and AC: ) Wherefore the
reafon of double the reAangie of BA and AC
to the triangle ABC is given.

Confir. Now let the angle C be fuppofed
acute, and from the point A let there be
drawn on CB the perpendicular AD.

Bemonftr. Foxafaiuch as the triangle CAB is
an Oxigonium triangle, and the perpendicular
AD being drawn, the fquare of CA and CB

E e 1 are

Digitized

4^8 EUCLIDE'j DATA.

t n, J. are equal t to the fquare of AB with twice
the reftangle of CB and CD ^ adding
therefore the common double reftangle of CA

and CB, the fquares
of CA and CB, with
the double reftangle
<3f the faid CA and
u 4. z, / \ \ CB9 that is to fay, u

/ I \ the alone fquare of

the line compounded
of ACB, are equal to
the fquare of AB,
T1 -w with the double of
J* the reftangle of CD
and CB, and over and above the double of
the rectangle of AC and CB, that is to fay,
the double of the reftangle contained under
the compound line of ACD and CB (for
s r. 2. the reftangle of ACD and CB is x equal to
the reaangles of AC and CB, and of CD and
CB :) Therefore the fquare of the line com-
pounded of ACB is greater than the fquare of
AC, by double the reftangle of ACD an£
CB. And feeing that the angle ACB is gi-
ven, and the angle BDA alfo given, the other
y 40 frop**u& CAD is given : Therefore y the trian-
gle CAD is given by kind, and therefore the
reafon of CD to CA is given, and by confe-
quence the reafon of the line compounded
2 6.fra$. of ACD to CA z is alfo given. Wherefore
the reafpn of the rectangle of thofe lines
a i, 6. compoynded of ACD and CB a to the red-
angle of AC and CB is alfo given. But the
reafon of the faid redangle of AC and CB
p 66.pvf>. to the triangle CAB I is given, feeing the
angle C is given ; therefore the reafon of
double the reftangle .of the line compounded
ACD and CB to the tfiangle CAB is given.

Digitized by Google

EUCLIDE'i

4? 9

Laftly, Let the an-
gle BAC be luppofed
to be obtufe, and ha-
ving prolonged BA
from the point let
the perpendicular CE
be drawn on the faid
line B \ prolonged ;
and let AF be pro-
pofed to be equal to AE,

Demovjtr. Foraimuch as the angle BAC is
obtufe, and the perpendicular CE being drawn,
the fquares of AB and AC, and the double
of the reftangle under BA and AE, or AF,
are all alike equal c to the fquare of BC, and c n«2,
adding the common double redangle of BA
and AC, the fquares of the faid AB and AC,
with the double of the rectangle of the fame
AB and AC, that is to fay, d the fquare of d 4. 2,
the line compounded of BAC, and the double;
of the reftangle of BA and AF are toge^
ther equal to the fquare of BC, with the
double of the reftangle of BA and AC. Let
the common double of the re&angle of BA
and AF be taken away, and there will remain
the fquare of the line compounded of BAC,
equal to the fquare of BC, with the re&an-
gle of AB and CF ; (for the reftangle of AB
and AC is equal e to the two rectangles of e 1, %%
AB and AE, and of AB and CF : ) Therefore
the fquare of the line compounded of BAC
is greater than the fquare of BC by the
double of the re&angle of AB and CF. And
fprafmuch as the angle BAC is given, the
angle CAE /is given. But the angle AEC isf x, 1Jm
alfo given; therefore the other angle ACE is
given : Wherefore g the triangle ACE is given g 40.^0/.
by kind, and therefore the reafqn of CA to
AE, that is to fay, to AF is given. There-
fore £the reafon of the faid CA .to FO b*S-frof.

E e.4 * 4, valid-

Digitized by Google

44* EUCLIDE* DATA.

etfo given. But the reafon of the fame CA
i 8ffrop. to CE is given ; therefore i the reafon of CE
to CF is alio given. Wherefore the reafon of
the reftangle of EC and AB to the redan-
gle of FC and AB is given; (for the reftan-
k 2. 6. gle is to the reaangle k as. CE is to CF) and
aifo that of the reftangle of AC and AB,
I 8. prop, to the reftangle of EC arid AB. Therefore /
the reafon of the reftangle of FC and AB to
the re&angle of AC and AB is given* But
the reafon of the reft angle of AC and AB
Vh66.prop.*o the' triangle ABC yt is given : Therefore
alfo the reaion of the double of the reAangle
of FC and AB, to the triangle ABC is gi-
ven. But the fame double of the re&angieof
FC and AB is that, whereof the {quite of the
Um compounded of BAC is greater than the
fquare of BC, whereof that fpace of which the
fqUare pf the line compounded of BAC is
greater than the fquare of BC, hath a given
rtafon to the triangle ABC.

OTHERWISE,
Conflr. Let the line B ' be prolonged to the
point D, ia fuch fort 1 AD may be equal to
AC, and let CD be drawn.

Demonjtr. Forafmuch as the angle BAC it
given, each of the angles ADC and ACD,
which is the half thereof fhaU be alfo given ;
and therefore the other angle DAC is alfo
U 40. prop* given : Therefore % the triangle ACD is gi-
ven by kind. Wherefore the reafon of AC
to CD is given. A»d forafmuch as the an.

gle ADC
is given \
J-et each
of the an-
gles DEC
and AFC
be made
\ equal to
the

i

Digitized by Google

EUGLIDFi DATA. 441

the laid ADC : Therefore feeing that the an-
gle BDC is equal to the angle DEC, and the
angle DBE is common to the triangles DBE
and DBC, the other angle BDE is equal to
the other angle BCD ; and therefore the tri-
angle BDE is equiangled to the triangle BDC.
Therefore 0 as EB is to BD, lb is BD to CB : o 4.6.
Wherefore the redangle of EB and CB, that
is to fay, p the rectangle of EC and CB, q p y.i.
with the fquare of CB is equal, r to the fquareq 5, 2.
of BD, that is to fay, to the fquare of ther 17.6.
line compounded of BAC > for AD is equal
to AC * and therefore the redangle of EC
and CB, with the fquare of CB» that is to
fay, the fquare of the line compounded of
BAC is greater than the fquare of the rz&~
angle of JBC and CE : I fay therefore that
the realbn of the faid reftangk of fiC and
CE to the triangle ABC is giv*n. Foraf-
much as the angle BDE is equal to the an-
gle BCD, and the angle ADC equal to the
angle ACD, the other angle CDE k equal
to die other angle ACB : Sit the angle DEC
is alfo equal to the angle AFC ; therefore
the remaining angle CAF is equal to the
remaining angle DCE- Wherefore the trian-
gle AFC is equiangled to the triangle DCE5
and therefore s as CA is to AF, £0 is CD s , ^
to CE ; and by permutation* as AC is to
CD, fo is AF to CE. Bat die reafon of AC
to CD is given : Therefore aifo the reafon
of AF to CE is given. Rom die point A
let AH be drawn perpendicular to BC : For-
plmuch as the angle AFC is givea, and the
angle AHF alio given, the third angle HAF
is *iv«n ; WheTefow t the triangle AHF isM°'F:<3
- given

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EUGLIDE's DATA.

given by
kind i
and by
confe-
quence
the rea-
fon of
AF to
AH is
given.

But the reafon of AF to CE is alfo given-:
Therefore u the reafon of AH to Ct is gi-
ven ; and theretore the reafon of the re&angle
of AH and BCarto the redan 2 le of BC and
CE is alio given. But the realon of the reft-
angle of AH and BC to the triangle ABC is
likewife given ; (for the re&angle is double
to the triangle) and the redtangle of BC and
CE is that whereof the fquare of the line
compounded of BAC is greater than the fquare.
of BC. Therefore that fpace of which thq;
fquare of the line compounded of BAC is

J greater than the fquare of BC by a given rea-
on to the triangle ADC.

Scholium.

■f The antient Interpreter pretending to Jbew
the conftruBion of the angle DEB equal to the
angle ADC, faith that on the line BD and in
the point D, the angle BDE ought to be made
equal to the ctngle BCD, and that the right lines
BC and DE be drawn until they inter feSt in E,
in fuch fort as he fuppofeth tbe angle BCD, ra
le given, but it is not.

The fame Interpreter afterward fiews how there
may univerfatly from a given point, be drawn a
right line given by pojition to a right line, making
an angle equal to a given angle. But we will
alfo re j eft this way, feeing we have elfcwhere

Jbewn

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EUCLIDE'/ DATA.

fljewn another more hrief and eafy. For example,
if we would from the point D draw t* the line EC
given hy p option a right line, making an angle
equal to a given angle ADC, as is here require^
we have no more to- do hut fo affume the point K
in the /aid line BC, and there make tie triangle
CKL equal to the given angle ADC : If the line
KL doth meet with the point D, it flail he the
lint ' required. But if it meet not with it, from
the point D let there he drawn the line DE, pa-
rallel to the faid KL,. cutting BC prolonged iu
E, and the angle t>EC Jball te equal to the given
angle ADC, for on the two parallel lines LK and
BE, there doth fall the line BE ; and therefore
the anile DEC z is equal to the angle LKC, which z
hath Teen made equal to the given angle ADC \
and hy conference the fame angle DEC is alfo
equal to ADC.

PROP, LXVIII.

If two parallelograms AB and CD, have to one
another a given reafon, and that a fide hath alfo
a given reafon to a fide, the other fide Jball have
likewife a given reafon to the*other fide.

Conftr. Let the reafon pf BE to FD be gi-
ven : I fay the reafon of AE to FC is alfo gi-
ven. For to the light line EB let there he
applied the parallelogram FH, equal to tho
parallelogram CD, and conftituted in fuch
fort as AE and EG may make one right line :
t Therefore KB and BH will alfp make one
fight line.

Demonftr.

444

1 1.6.

b 14. 6.

c ij

4 *9* !•

EUCLIDE'* JD^r^.

«. •

Demonjtr.
Forafimich as
the reafon of
AB to CD is
given , and
chat EH is e-
qual to the
laid CD ; the
T-T reafon of AB
T JJ to EHisgi.

ven 5 and

therefore the reafon of AE to EG is alfo gi-
ven. Seeing therefore that EH is equal and
equiangled to CD, as h EB is to FD, fo is
FC to EG. But the reafon of EB. to FD is
given : Therefore alfo the reafon of FC to
EG is given. But the reafon of AE to the
fame E(5 is alfo given : Therefore the reaCon
of AE to FC is given.

Scholium.

t EUCLIDE having pofited AE and EG di-
reftly in one right fine, prefently concludttb that
KB and BH JbaU alfo make a right line ; ha
wtjhafl demonftrate it thus. Seeing tie lines JE
and EG ate pitted iireSly, the angles AEB and
BEG c are equal to two right angles ; and ferine
that AB is a pataUelogram, the Una AK and EB
are parallels, on which the line AE doti fall ;
and therefore the two internal angles A and SEA
d arc alfo equal to two right angles* and taking
away toe common angle SEA, tlxrt will remain
the angle A, equal to the angle BEG $ and eon*
fequent'ty their oppofite angles EBR and H are
aifo equal to one another : Again, feeing that BG
is a parallelogram, the two lines BE and HG are
parallels, on which BH doth fall ; and therefore the
two inttrnal angles If and EBH d are equal ta
two right angles. But it bath been demonftratcd

that

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EUCLIDF/ DATA. 44y

that H is equal to EBK : Therefore the two an-
gles EBK and EBH are alfo equal to two right
angles i and therefore c the two lines KB ande 14.1.
BH do meet dircBly according to EUCLID &

OTHERWISE,

Covftr. Let the given right line K be expo-
fed, and feeing that the realon of A to £ is gi-"
ven, let the fame be made of K to lL 5 there-
fore the reafon of K to L is alfo given.

Demonjh. But K is given ; therefore/ L isf z. prop.
alfo given* Again, feeing that the reafon of
CD to EF is given, let the fame be made of
K to M : Therefore the reafon of K to M is
given. But K is given, therefore,? M is alfog l* prop*

given ^ and
therefore the
reafon of L ta
M is given.
Now feeing that
A is eouiangled
to B, ithe rea-h 2?, &
fon of the faid
A to B is corn-
pounded of that
of the (ides,that
is to fay, of CD
to EF, and of CO to EH. But alfo the reafon
of K to L is compounded of K to M, and of
M to L ; therefore the reafon compounded of
CD to EF, and of CG to EH, is the fame
With that which is compounded of K to M,
and of M to L (the reafon of K to L being
the fame as of A to B : ) But the reafon of
CD to EF is the fame as of K to M : There-
fore the other reafon of CG to EH is alfo the
fame as of M to L. But the faid reafon of
M to L is given : Therefore alfg the reafon
of CG to EH is given.

PROP.

* >

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446

EUCLIDE'i DATAi
PROP. LXIX.

If two fa-
raUelogramSj
' CBandEH,
having the
angles D and
Fgiven, and
that a fide
hath alfo a
given reafon

to a fide i in like manner the otlHA fide Jb all have
a given reafon to the other fide.

Conjlr. Let the reafon of BD to FH be alfo
given : I fay that the reafon of AB to EF is
given. For if CB be equiangled to HE, it is
manifeft by the precedent Propofition ; but if
it be not equiangled thereto, let the right
line DB be conftituted, and in the given point
B therein, let the angle DBK be made equal
to the angle EFH, and finilh the parallelo-
gram DK.

Demonjtr* Forafmuch as each of the angles
BKL and BAK is given, f the othpr angle

a 40. frof . KBA is given : Wherefore the triangle a ABK
is given by kind ; and therefore the reafon of
AB to BK is given. But the reafon of CB to

b 55-?*>P« EH is fuppofed to be given, and h CB is equal
to DK ; therefore the reafon of DK to EH
is given $ and feeing that DK is equiangled to
EH, and the reafon of the faid DK to EH is

c 6B.frop. given, as alfo that of DB to FH, c the reafon
of BK to FE is given. But the reafon of the

d 29. 1, faid BK to BA is alfo given : Therefore d the
reafon of AB to FE is given*

Scholium.

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EUCLIDE'* J>ATA. 447

Scholium.

f EUCLIDE fupporeth there that a paraUehh
gram having one angle given, all the otljer an~
gles are alfo given} ana as well the antient Inter-
preters as others, do give the reajons why, the an-
gle F being given, the other angle E Jball be alfo
given, it being the remainder of two right angles,
for that' on the parallel lines EG ana FH were
doth fall the line EF, which makes e the two in-
ternal angles (of the fame part) F and G, equal e 29. t*
to two right angles. But to thofe angles f the op-i J4* U
pofite angles G and If are equal, and therefore they
are alfo given.

From whence it follows that the angles BBC and
F being given by fuppofition, all the other angles
of the two parallelograms CB and EH, are alfo
given : Therefore the angle DBK having been made
equal to the angle F, the angle K Jball be equal
to the angle E, and given as that is : But the
angle BAL which is oppofite to the given angle
BDC, is alfo given \ and therefore BAK which is
the remainder of two right angles, Jball be alfo gi~
ven 1 in fuch fort as in the triangle ABK, the two
angles BAK and BKA are given, as EUCLIDE
doth declare in this place.

PROP* LXX.

If of two parallelograms AB and EH, the fides
about the equal angles, or a\out the unequal an-
gles (yet never thelefs given angles have to one
another a given reafon, to wit (AC to EF, and
CB to FH) alfo the fame parallelograms AB and
EH Jball have to one another a given reafon.

Conjlr. For let AB be prolonged to EH, and
on the right line CB let the parallelogram CM
be applied equal to the parallelogram EH, in

fuch

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448

EUCLIDE'i DATA.

fuch fort as AC may be direft to CN ; that
is to fay, that AG* and CN make one right
a fch. 68.H°e > and b/ confequence DB fhall be a direft-
trop. w^r^

£ 1 Vemonjlr. Forafmuch then as CM is.equi*

angled and equal to EH, the fides about the

b 14. 6* equal angles (hall , be reciprocally b proportion
nal ; Wherefore .as BC is to HF, fo is FE to

NC. But

HF is gi-
ven :

Therefor*
the reafon

of FE to
NC is ah
fo given.

But the reafon of AC to the lame Er is gi*
c S.prop. yen : Therefore c the reafon of AC to NC is
alio giveq. Wherefore the reafon of AB to
d i«6. CM is given ; (for it as the fame d as of AC
to CN J But CM is equal to EH : Therefore
the reafon of AB to EH is given*

Conftr. Now fuppoie AB not to be equian-
gled to EH, and on the right line CB, and
in the given point C therein : Let there he
conftituted the angle BCK, equal to the gi-
ven angle F, and fo finifli the parallelogram
CL.

Vemonjlr. Forafmuch as the angle ACB is
given, and the angle BCK alfo given, the re-
maining angle ACK is given :* Therefore the-
fe 4a/r^« triangle ACK t is given by kind ; and there-
fore the reafon of AC to CK is given : But the
reafon of AC to EF is alfo given : Therefore
the reafon of CK to EF is given. But the
realbn of BC to HF is alfo given, and
the angle BCK is equal to the angle F \ there-
fore

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EUCLIDE'/ DATA 449

fore (by the firfl: part of this Proportion) the
reafon of CL to EH Is given. But to the faid
CL, AB is equal : Therefore the reafon of
AB to EH is given.

PROP. LXXL

If of two
trianglts ABC
md DEF, she
fides about the
equal angle;
A anS D$ or %
elfe about the
unequal angles
{yet ncverthe*
lefs given angles) have to one another a given rea-
fon (to wit AB to DE and AC to DF) the fame
triangles /baU have alfo to one another a given
reafon ABC to DEP.

Confin Let the parallelogram AG and DH
be finifhe^.

t)cmonftr. Seeing that the two parallelograms
AG and DH, have the fides about the equal
angles A and D, or elfe about the unequal
angles (neverthelefs given) have a given rea-
fon to one another, the reafon a of the para!- a jo.prop.
lelogram AG to the parallelogram DH is gi-
ven. But the triangle ABC is the half of
the parallelogram AG b and the triangle DEF b
the half of the parallelogram DH. . Therefore
the reafon of the triangle ABC to the trian-
gle DEF is given.

F f

PJIOP

«

is

»

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4yo

EUCLIDE', DATA.

I ►

PROP, LXXIL

t If of two triangles ABC and DEF, the hafts
MC and EF% are in a given reafon, EC to JEF9
and that from the angles A and D, there be
drawn to thofe hafes the right lines AG and DH,
making the equal angles AGC and DHF, or elfe
unequal (yet neverwelefs given) which Jball have
to 'one another given reafons AG to DH, thofe
triangles ABC and DEF Jball have alfo a given
reafon to one another, to wit, ABC to DEF.

Conflr. For let the parallelogram KC and
LF be finifbed.

Demonjtr. Forafmuch as the angles AQC and
DHF are equal, or unequal (yet given) and

a 29. I. the angle AGC a is ec,ual to the angle

KBC. But the angle DHF equal to the an-
gle LEF, the angles at the points B and E
are equal, or elfe unequal (yet given) and for
that the reafon of AG to DH is given, and
AG is equal to KB. But DH is equal to LE,
alio the reafon of KB to LE is given. But
the reafon of BC to liF is alfo given, and
the angles at the points B and E are equal, or

fc7G.fr0f.elfe unequal (yet given :) Therefore b the rea-
fon of the parallelogram KC to the parallelo-
gram LF is given ; and therefore the reafon of
the triangle ABC to the triangle DEF is gi-

€41.1* ven, feeing thofe triangles care the one half
of the parallelograms.

PROP-

\

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EUCLIDE'/ t>AtA.

■

*■

PROP. LXXIIL

4f*

I/of
/wo pa*
jH rallclo-
gramsA-
B and
EG,

J*

equal an-
gles C and F, or e^i? a lout the unequal angles {but
nevertbelefs given)are in fitch fort to one another j/that
as the fide CB of the fir fi, is to the He FG of the
fccond ; fo tlx otter fide EF of the frond, is to
fome other right line CN. But that the other fide
AC bath alfo to tlx fame right line CN a given
reafon, thofe parallelograms will have alfo to one
another a given redfon A? to EG.

Cohftn For in the firft place, let the paral-
lelogram AB be cquiangfed to EG, and ha-
ving placed CN dire&ly to AC : Let the pa-
rallelogram CM be finifhed.

Demonftr. Forafmuch then as CB or ifM it*
e^ual, is to FG, fo is EF to CN, and that
the angles N and F are equal (for N is Cqual
to the angle ACB, which is put equal to F)
the parallelograms CM and* EG a are equal ; a 14$ 6.
But as AC to CN, fo * the parallelogram AB b 1. 61
is to the parallelogram CM or EG : , Therefore
feeing that the reafon of AC to CN is given,
the reafon of AB to EG is alfo given.

Conftr. 2; Now fuppofe the parallelogram
AB not to be equiangled to the parallelogram
EG, and let there be conftituted at/the given
point C in the line GB, the angle BCK, equal
to the angle EFG, and lb finifn the parallelo*
gram CL*

f f 2 Utmonjln

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4T*- EUCL1DE V VUKAk

Deinonjlr. i. Seeing that each of the angles
ACB and KCFIs given, the remaining angle
cfcb.6$. ACK is alio given. But c the angle CAK is
prop. livtt, as alfo the remaining angle AKC :
d 4°'F^- Therefore d the triangle ACK is gfeeiF.by
kind$ and thctefore the reafon of AG to *CK
iV given. But the reafon of the fame A0 to
e 8. prop. CN is alfo given : Therefoie e the. reafort of
CK to CN is given. ' And feetrrg That as CB
is to FG ; fo is EF to the right line CN* to
tfhieh the other fide KC hath a given rear-
fon* and that the angle BCK is equal to the
ingle F* \he reafon 6f the parallelogram CL»
to the parallelograih EG is given (by the firft
part of this Proftefition) but the parallelogram
CL is equal td the parallelogram AB: There^
fere the reafon of the parallelogram AB t*
th« parallelogram EG is given.

PROP. LXXIV.
It two parallelograms {as in the former figure)
. JtB and EGy in Coital angles € ttild t\ or tlfe in
iftltqttal tuigtes (ytt neverthelefs given angles)
have a given reafon to one another^ us one fide CM
of the pft Jhall $% to one fide FG of the fecdnd9
fo the Otijer fide EF of the fecond, IbaH be to that
fo tin which the other fide AC of the firft hmtb* a
given reafon.

Conflr. For eith'er AB is equiangled or not 9
fuppofe it iri the firft place to be equi angled,
and to the right line BG kt there be
applied the parallelogram CM* equal to the
parallelogram EG, and fo ppfited» as that
a /<:/;. 68. AC and CN maybe direct Therefore a DB
pop. aitd BM fcaH be alfo direct (that is as orte
•right line*) \ , »

Demohfir. Seeirrg that 'the itfafoto of AB to
EG ls given^ and that CM is equal to £(3,
the realon of AB to CM is* alfo given and
-tKeretii the realon of AC to CN is given
^ (ieeing

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i

EUCLIDE'x DJ7A. 45 ;

{fceing AB is tp (3M, b as A£ is to CM 0 *adb 1. 6.
for that CM is equal and equiaiigled to EG,
the fides about the equal angle* $f the paral-
lelograms CM andEQj f are reciprocally pro c 14. 6.
portional 5 and therefore as CB is to FU, fa
is EF to ON. But the reafon. of AC to CN
is given : Therefore as CB is to FG, fp ks
EF to that to w^ich AC hath a given reafon. -
Conjh. 2. Now fuppqfe AB not tp be cquir
angled to EG, and in the given point C of
the line CB, let tjiere be conftiitwd the:
rjeBOK tqual to the angle EFG, and finiffi
he parallelogram CL. ; j V * \ I 1

■Demoiift . a. Seeing thea tlaat the reaCoo of
AB to EG is given, and d that AB is equald 56. r.
to CL, alfo th« reafon of CL. to EG is given,
an£ the angle BCK is equal to the angle Ff
*nd therefore' CL > is equiangled \£ IG:efcb.6g.
Therefore ft>y the firft. part of this Trdpofi. prop
tiori) as CB is to FG, fo is EF to^ttyt to
Che wjiich CXfrath a, given reaionJ* Jfa the
teafon of AO-to CK. wUgiven ; (is ab^ars
by what harli been demonfhated, in th* lfcter
part of the precedent Propofitio/i.) Thtrefcre
3s CB is V© FQ, fo is EF to /hat to frhi\h
AG hath a given reafoiu / , _\

e " ' * I — *

P R O P. ~LXXV.

• m n it * * •

* :yZ \. • / >%c . <wd DBF, in

equal augks

elfe unequal
{yet nevertbe-
lefs given)

9 . have to one

4v$tlm 4 given reafonn <v thefM AB of thepft%
fiall be to the fide DE of the ficoni, fo the

F r j ' other

"6

Digitized

4j4 EUCLIDE'i DATA.

other fib VF of the fccond, JbaU h to that
r right line to the which the other fide AC of t1*
fttft hath a given reafon.

Conftr. For let the parallelograms AG and
DH be finifhed.

Demonftr. Forafmuch as the reafon of the tri-
angle ABC to the triangle DEF is given, air
fo the reafon of the parallelogram AG to the
parallelogram DH is given.

Seeing therefore that the two paraUelogran*
AG ana E>H in equal angles, or unequal anr
gles (neverthelefs given) have to one another
1 74 pof* a given reafon ; as a AB is to DE, fo is DF
to that to which AC hath a given reafpn.

PROP, LXXVIt

If from the top A of a
triangle ABCy given fo kind,
there be drawn to the }afe
BC, a perpendicular line AD9
that line AD fit all have to
tU hafe BC a given reaforu
Demnftr. For feeing that
the triangle ABC is given
by kind, the reafop of AB
to BC is given ; jand the
angle B is ajfo given. But
the angle ADB is given ; therefore the other
a 40. /if/, angle BAD is given, wfcerefpre a the trian
g!e ABD is given by Jrind * and therefore the
feafon of AB to AI> is given. But jhe rea-
1> &$rop. fon of AB to BC i*jgiven : Therefore h trij rea
fon of AD to fiC is given.

■

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EUCLIDE'i J>ATA.

4ff

PROP. LXXVIL

If two figures JBC and l)EF9 giyen h kind,

have to another a given reafon, the retfon alfo

fiall be given of which you pleafe of the fides of one

of the figures, to which you fleafe of the fides of

the other figure. \f* f ^ i

■ /y Conftr. For

on the liglir
lines BC **#
EF, let thire
be defcnbjpd
the fquares UXj

and EH. I
Jkmovfir.Fbt-

armuch as on
one and the
lame ught line
BC, aiedefcri-
bed two figures
ABC and BG. „ _
given by kind, * the reafon * the ftid ABC M> H»
u> BG is given . In like manner, the reafon of
DEF to EH is given ; and feeing that the tea.
fon of ABC to 1)EF is given, and alio that
of the fame figure ABC to BG ; and a*ain the
reafon of DEF to EH : * the realon or BG b B-p-Qf*
tp EH is given * and therefore the realon of
BC to EF is alfo given.

t .

F f 4

PROP.

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EUGLlDEV bAtAt

PRO?. tXXVitt.

-

■yew
E m/bs

gure DFr

y ant that
one ' .fide

BC bai% a
given rea*
fon to fife
fide D£,
the reftin*

gkd figure Z>F is given ly fcfiuj.

Confir. For 0n the right Jine.BC. let jfee
fquare BH be defcribed, and to; the right line
■ t>E, let the parailelogram DK be applied equal
to BH, in fuch manner, as that QD and Dl

a [ch. 68. may be placed dire&ly, a and by confequence

pop. FE and JEK alfo direttly,

Demonjlr. Therefore feeing that on one and
the fame right line BC are defcribed the two
re£liline figures ABC and BH? given by kind,

bty.pof.h the reafon of ABC to BH is given. But
/ the reafoir of the faid ABC to DF is alfo

c 8. prop, given : -Therefore c the reafon of BH to DF
is given. But BH is equal to DK : Therefore
the reafon of DK to DF is alfo given. And
feeing that BH is equal and equiangled to
DK, both the one and the other being reft-

d 14. 6. angles, d the fides of thole figures are recipro.

cally proportional ; and as BC is to DE, fq
is Dl to CH, Eut by fuppofition, , the reafon
of BC to DE is given ; theiefore alfo the
reafon of Dl to CH is given ; but the reafon

9f

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EUCLIDE'/ DATA.

of DI tQ DO is alTo given I (for DI is to
DG e as DK to DF:) Therefore / riiq fea^c r. 6.
fon of DG to CH is given. Buf.CHij equal 8, prop*
to HC, feeing that BH is a fyiare ; therefore
the reafon 01 BC to DQ is given. But the
reafoft of the fame JJC to DE is alfo given ;
therefore the reafon of DE to DO ie given,
and the angfe at D is a tight angle ; Tht
fore g DF is given by kind.

1010-

J

* prop, r.unlro . tr0*

7f two triangles ABC and EFQ, Lave an an-
gle • B equal to an angle F. But front the equal
angles S and F there be drawn perfendieulars BD
and FHy to the hates AC and EG $. and that a*
the bafe AC of the firft triangle ABQy is to the
perpendicular Buy fa aljo the bafe EQ of the other
triangle EFGyv to the perpendicular FM, thofe
Wangles ABC and EFG are $ytian$lel

' Conftr. For about the tri angle EFO let there
be defcribed t]ie tire]?? FLO, then on the right
line EG, and fa) the point E given therein,
let there be made the angle GEL, equal to
the angle C, and let FL and LG be drawn,
and the perpendicular LM% 1

tkmynfiik

« •

V

t »

Digitized by

4y8 EUCLIDE's DATA.

Deinonjtr. Seeing then that the angle GEL
- is equal to the angle C, and the angle ELG
»%f*l« is equal tt> the angle EFG, a they being in
one and the fame legment of the circle 3 the
third angle EGL is equal to the third angle
A* Wherefore the triangle ABC is alike to
the triangle ELG, and the perpendiculars BD
and 131 are drawn : Therefore *f as AC is
to BD, fo is EG to LM ; but by fuppofition
as AC is to BD ; fo is EG to FH : Therefore
K, r HM is equal to FH. But the faid LM
J Ji , is c parallel to FH : Therefore d FL is alfo pa-
raMtoEG; and therefore the angle FLE

m il9 V * *» to the anSle LEG- Bl* tbe a<*-

gk C is alfo equal to ihe faid angle LEG, and

f zr. z. roe angle FLE to the angle FGE f : Therefore

alfo the angle C is equal to the angle FG E.

But by fuppofition the angle ABC is equal

to the angle EFG : Therefore the third an*

gle BAC is equal to the third angle FEG :

Wherefore the triangle ABC is equiangled to

t the triangle EFG,

Scholium. /V

f Now that as AC is to BD, fo EQ is to LM,
it is by fome thus demonjlrated. For/fmuch as the
angle C is equal to the angle GEL^and the, angle
BuC to the angle LMEy each being a right an-
gle, the other angle CBD is equal to the other

g 4. 6. angle ELM : Therefore g as EJtt'U to ML, fo is
CD to DB. Again, feeing the angle ABC is*eqital
to the angle ELG, and the angle CBD to the
angle ELM, the remaining angle JBD is equal
to the remaining angle MLG , but the angle
JDB is alfo equal to the angle LMG } and tfyre-
fene the third angle J is equal to the third angle

h 4. 6. LGM: Therefore h as AD is to DB, fo is GM
to A(L. But it hath been demonjlrated that as

; CD

Digitized by Google

EUCLIDF* DJTJ.

4f»

CD is to DB9 fo is EM to JUL ; . Therefore i asi 14, 5.
JCu/o BD,fo is EQ to hid.

PROP. LXXX.

If a triangle ABC hath one angle A given, ami
that the reB angle contained under the fides AB and
AC, comftifing the given angfc Ay ktill^ a given
reafon to the fquare of tlx other fide JC, the
triangle ABC is given by kind.

Conjtr. For from the points A *nd B, let
there be drawn the perpendiculars AD and
BE.

Demonftr. Forafmuch as the angle BAE is
given, and alfo the angle AEB, the triangle
ABE is given by a kirid ; and therefore the a 40. prof,
reafon otAB to BE is given : Therefore the •
reafon of the reftangle of AB and AC to the
reftangle of BE and AC is alfo given (for

it is the fame reafon? 1*6*
b as of AB to BE.) But
the reftangle of AC and
BE is equal to the red-
angle of BC and AD;
for that each ofthofe red-
angles is c double to thec ^f.i.
triangle ABC. Therefore *
the reafon of the rectangle
of AB and AC to the red-
angle of BC and AD is
*lfo given. Bit th$ reafon of the reftangle of
AB and AC to the fquare of BC is given :
Therefore d alfo the reafon of the reftangle of 4&frop.
" BC and AD to the fquare of BC is given ;
and therefore the reafon of the right line BC
to the right line AD is given. (For that c thee i^tf.
reftangle is to the fquaie as AD to BC.) Now
let the right line FG given by polition and

Sagnitude, be expofed 5 and thereon let there
i defoibed the ftgment of a circle FIG *

Y capable

!>J7»

Digitized by

magnitude. But it is alto given by poiiti
andthe point G is given : Therefore the po

<4<5p EUCLIDE'i flUHT-A

^ .... capable of an angte equa.1 to the angle A*
And feeing the faid angle A is given, ajfp the
angle in the fegment FLG fnall be given ;
fQ.def. and therefore /the fame fegrjjent is given by
pofition. From the point G let there be ereft-
td at right angles on the line FG, the line
g 4* ' which g is giT

Tfc ven by pofition : Let
it be. fo made, that as
BC is to AD, fo FG
may be. te GH j and
feeing that the reafon
of BC to AD is gi veil,
alio that of FU. to
^ GH is. given. But
FG is given : There-
fore h GH is given by

uion,

ppjnr

i 2.7, prop.H is i alfo given. Now by the point H let
there be drawn HI, parallel to FG, and that
k 2&\p>-o/\line HI fliali be given by k poiiti©n. But the
Xeg'ment, of the circle FIG isajfo given by
1 2$. pofition. Therefore I the point I is given.

Let the right lines IF and IG be, drawn, and
the perpendicular IK ; Therefore IK is given
by pohtion. Bur tjie point I is given, ^s aifo
m 16.pr0p.each of the points F and G : Th^e^ef^i
each of the lines FG, FI,^4ind IG is giUii
n 39.pop.ty pafition and magnitude : Wheiefore n the
triangle FIG is givrn by kind 5 and feeing
that as BC is to AE, (b is FG to GH, and
o i^P'of.p that to GH, IK is equal, as BC is to AE,
fo is FG to IK, and the angle A is equal to
ff9.pr0p.pai angle FIG: Therefore/) the triangle ABC
is .equiangled to the triangle FIG. Bui FIG
is given by kind: Therefore alio the triangle
ABC is given by kind.

QTHEJU

.1

by Google

EUCLIDE'x DATA,

6.T HE RWISE

468

Conjlr. Let the trfeflgle ABC, Urhofe angle A
is given, and the realon of the reftangle con-
tained under AB ami AC, to the fquare of BC
be given : I fay that* the triangle ABC is gi-
ven by kind. . /

Demovfit. For feeing the angle A is given,
that fpace of which the fqUare of the line
compounded of BAC is greater than the fquare
of BC, y hath a given reafon to the triangle q 67. prop.
AfcC« Now let that fpace be D: Therefore
the reafon of D to the triangle ABC is given.
But the reafon .of the triangle ABC to the
iedangle of AB and AC is given $ r feeing r 66. prof.

the angle A

Therefore nhe s 8. prop.
reafon of the
fpace D to the
reftangle of
AB and AG is
given. But the
reafon of the
re&angle of
AB and AC

L

• 1

• < ■

to the fquare of BC is alfo given : Therefore
s the reafon of the fpace D to the fquare of
BC is given* Wherefore by compounding, t
the realon of the fpace D, with the fquare of*
BC to the faid fquare of BG is given : There-
fore the reafon of the fquare ot the line com-
pounded of BAC, t<? the fquare of BC is gi-
ven ; (for that the fpace D with the fquare of
BC is equal to the fquare of the line cam-
pounded of BAC j) a«d therefore u the feafon ufcL.^u
of rhe faid 4ine compounded of BAC to hGprop. 1
rs given. But the angle A is alfo given.:

There*-

Digitized

jfil EUCLIDF/ DATA,

% 4$. pap. Therefore * the triangle ABC is given by
kind*

PROP* LXXXI.

i

A

> *

D

^^^^^ ^^^B^

B

•

E

■

C

F

If of three right Met
A, B% and C, proportion
nal to three othtr pro^
portional right lines D%
E, and Fj the ^extremes
A and D, C jwi F, art
in a giveft tea/on (to
wit \ as A to D, and C to F,) alfo the means £
and E /ball he in a given reafon, and if one ex-
treme hath a given reafon to an extreme^ and the
mean to the mean> the other will have alfo a gi-
ven reafon to the other.
Demonftr. Foiafiquch as the reafon of A to

D, and of C to F is given, the reftangle of A
a 70. pop. and D a {hall have a given reafon to the red-
angle of C and F. But the rectangle of A and

b 17. 6. D is e4u*l * to the fquare of B ; and the redan •
gle of C and F to the lquare of £. Therefore
the reafon of the fquare of B to the fquare of

c/ck$i. E is given; and therefore t the reafon of the

pop. line B to the line E is alfo given.

Again, Let the reafon 01 A to D, and B to

E, be given : I fay that the reafon of C to F
is alfo given. For feeing that the reafon of A
to D, and of B to E is given, alfo the reafon

d yxpop. of the fquare of B d to the fquare of E is gi-
ven. But the fquare of B is equal to the reft-
angle of A and C, and the fquare of E to the
Tedangle of D and F : Therefore the reafon of
the re&angle of A and C to the reftangle of D
and F is given. But the reafofl of a fide A

. »k ft . to a fide D is given : Therefore e the reafon of ,
eoo.p^the othcf fid(j c tothe Qth^ fide F h alfo

given.

PROP-

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EUCLIDE'/ DATA. 46 }

PROP. LXXXII.

1

A If there be four right lines A,ByCi

B — — andD, proportional, as the firjjt A%
C ■ ■ JbaU be to that tine to which the fe-

ll — cond B hath a given rcafon, fo the

E third C, JbaU be to that to which .

F — the fourth D hath A given reafon.

Conftr. Let E be the line to
which B hath a given reafon, and let it be fo
as that B may be to E, as D is to F.

Demonjtr. Now the reafon of B to E is given,
therefore alfo the reafon of D to F is given.
And feeing that as A is to B, fo is C to D.
And again, as B is to E, fo is D to F, by rea-
fon of equality as A is to E, fo is C to F..
But E is that line to which B hath a given
reafon, and F that to which D alfo hath a gi-
ven reafon: Therefore as A is to tfcat to whicIV
B hath a given reafon, fo C is to that to which
D hath a given reafon.

PROP. LXXXIH.

*

A If four right lines A%

•and D, are in fuch fort to one

B another , that of any time of th

A% By and C, and a fourth ■
taken proportional, to which that
line Dy which remains of tlx four
P lines, bath a given reafon, she

• '■■ four lines A, n, C, and E, are ■
E proportional \ as the fourth D is

to the third C% fo tlje fecond B

JbaU be to that so which the firjt

A hath a given reafon.

Demwjlr. Forafinuch as A is to as U is to
E, the re&angle coutained under A and E a is*
equal to the re&angle contained under B and

Digitized by Google

4«4

EUCLIDEY -Qjitok

b I.&

C ; and feeing that the reafon of D to E is
given, alfo fhall be 21 veil the itafon of the
reftangle of A and D to the rectangle of A and
~ (for h it is the fame reafon as of D to E^)
it the re&angle of A and £ is equal to the
rectangle of B and C : Therefore the .reafon of
the rectangle of A and D to the reftangle of B
c arid C is given. Wherefore c as D is to C, lb

is £ to that to*\Vhich A hath a given reafon.

P R 0 P. LXXXIV. • • •

If two right lines AS
and AE comprehendh^g
a given Jface JJP in 4
given angle BA£, and
that the one AS he
greater thin the other
AE hj a given line CB9
alfo eaeh of the lines
AS and AE is given.
Dcmonjlr. For feeing that AB is greater than
AE by the given line CB, the remainder AC is
equal to Ari : Finifh the parallelogram AD.
Therefore feeing that At is equal to AC, the
, feafon of. AE to AC is given, and the angle

a fch 6u A is alfo given : Therefore a AD is given by
pof.l kind. Wherefore the giv*n fpace AF is applied

• to the given right line CB, exceeding it by the
given figure AD given by kind 5 and therefore

bsp.pof .b the b/eadth of the excefs is given. There-
•fore, AG is given. But CB is alfo g iven: There-

• fore the whole AB is given. But AE is alfo
^iten : Therefore each of the right lines AB
and AE is given.

c.1

PROP.

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EUCLIDE'i DATA:

PROP. LXXXV.

4< $

If two right lines AC and
CD, dp comprehend a given
Jpjce AD in a given angle
ACDj the line compounded
of thofe lines AC and CD
is given, alfo euh of thofe
lines AC and CD is given.
Conftr. For let AC be
prolonged to the point B, and Jet CB be put
equal to CP, then by the point B let BF be
drawn parallel to CD, and fo finifli the paral-
lelogram CF.

Demonjir. Seeing then that CB is eaual to
CD, and the angte DCB is given ; tor that
angle that follows is the given angle ; and
therefore a the parallelogram DB is given bya rrj#g|.
kind ; and again, feeing that the Tine com- pr0p*
pounded of ACD is given, and CB is equal to
CD, alfo AB is given. And thuf to the right %
line AB there is applied the given fpace AD,
deficient by the figure DB given by kind ; and
therefore b the breadths of the defefts are al-b fafrof.
fo given : Therefore the right lines DC and CB
are given. But the compounded line ACD is c A.pW.
*lfo given : Therefore c each of the lines AC
and CD is

PROP.

Digitized by Google

466 EUCLIDE'j TtJTX.

FKOF. I2HSXVK

if right lines AB
and BC, do compehend a
given Jpace AC^in a given
angle ABCy the fquare of
the one BC> u greater
tlmn the fquare of the
otlkr ABy by a given
fiace (vet in a given rea-
fon) gtf& ebfo «f thofe
lints IB &nd BC /ball be given. '

Demonfir. For feeing that the fquare of BC
is greater than the fquare of AB by a given
{pace (yet in a certain reafoa.) Let the given
(pace be taken away, that *s to fay, the reft-
angle contained under CB and BE : Therefore
z ii.dsf. a the reafon of the remainder, b which is the
b i. a. reftangle contained under BC a*4 CE to the
(quart of AB is giveth And ferafftiuch as the
rtftangte t and BC is given, and

c I. prop, alio that of CB and BE, their c reafoft fejgivefk
But as the reftangle under AB and BC is to
d r. 6. the reflangJe under CB and EB, d fo AB is to
BE £ aid therefore the reafon of AB to BE
e jo.prop is given : Wherefore e the reafon of the fquare
of AB to the fquare of BE rsalfo grvein But
the reafon of the fquare of AB to the reftan-
f &prof. gle under BC and CE is given : Therefore /
alfo the reafon of the reftangle under BC ana
CE to the fquare of BE is given. Wherefore
the reafon of four tiJbes the re&angle under
BC and CE to the fquare of BE is given >
g &Prop. and by compoundiag, g the reafon of four
times the redangfe under BC and CE, with
the fquare of BE to the fquare of BE is given.
• But four times the reftangle of BC and CE,
h 8. i. with the fquare of BE, h is the fquare of the
compound line BCE : Therefore the reafon of

the

Digitized ky Google

EUCUDW, DATA. 4*7

Iquarc of the compound line BCE to the fquare
of BE is given : wherefore i the reafon of thei $4. prop.
line compounded of BC and Cfc to BE is gi-
ven, and by compounding, k the reafon ot the k 6. prop.
compound of the lines BC, CE, and BE, that
is to lay, the double of BC to BE is given ;
and therefore the reafon of the only line BC
to BE is altb given. But as BC is to BE, /foi 1. 6.
the re&angle under BC and BE is to the fquare
of BE ; Therefore the reafon of the rectangle
Under BC and BE to the fquare of BE is gi-
ven But the reftangle of BC and BE is gi-
ven ; Therefore m the fquare of BE is alfo gi- m i.prop>
ven, and confequently the line BE is given.
Wherefore BC is alio given, feeing thar the
reafon of BE to BC is given. But the fpace
AC is given, and alfo the angle B : Therefore
* AB is given* Wherefore each of the lines n $7. pop,
A B and BC is given.

Scholium.

t Inflead of faying in this place [what is un-
der, &c.\ we have ufed this Word Jtettangle, it
liing manifeft by what follows that fitch was tljt
intention of EUCLID E, feeing he makes ufe rntht
faid Demon/It ation of the fecond and eighth Pro-
pfition of the twelfth Element \ and alfo that
the fpace or ParaUelogram riven being not rectan-
gied> it may he reduced thereto, making on BC9
and in the given point £, a right angle CBJy fo
4U that there will be two Parallelograms conflic-
ted on one and the fame lafe £C, and between the
fame par all els, as in the 69 th Prop ofit ion by meant
whereof this Conclufion is drawn.

Note, This ferves alfo for the next Prop.

xigt prop,

Digitized by

4*3

EUCLIDE'/ DATA.

PROP. LXXXVU.
A

ftwo right lines JB
BC, do comprehend
a given Jpace AC, in a
given angle B, the fquare
of the one BC, is greater
than the fquare of the
other AB, by a given Jpace,
tlfo each of thofe Urns JB and BC Jball he
given*

Demonftr. For feeing that the fquare of BC
is greater than the fquare of AB by a given
fpace : Let the given fpace be taken away, and
let the reftangle be contained . under BC and
a 2. z. EE : Theiefoie the remainder, a which is the
re&angle of BC and CE, is equal to the fquare
of AB. And feeing that the re&angle of BC
and BE is given, and alfo the fpace or reftan-
cle AC ; the reafon of the faid re&angle of
b i. 6. SC and BE to AC is given. But as b the reel-
angle of BC and BE is to the reflangle of
:AB and EC, fo is BE to AB : Therefore the
reafon of BE to AB is given, and therefore
c %o.prop& the reafon of the fquare of the faid DE to
the fquare of AB is alfo given. But to that
•fquare of AB the redangle of BC and CE is
equal : Therefore the reafon of the laid reftan-
gle of BC and CE to the fquare of BE \^ gi-
ven ; and therefore the reafon of the quadruple
of the faid reftangle of BC and CE to the
fquare of BE is alfo given j aud by compound-
ing, d the reafon of four times the rectangle
of BC and CE, with the fquare of BE, to trie
faid fquaie of BE is given. But four times
the re&angie of BC and CE, wiih the fquare

<c 8. 2. °^ ^» e IS r^e f(luare of the compound line

Therefore the reafon of the fquare of
that compound line BCE to the fquare^f BE

is

d 6. prop.

Digitized by

EUCLIDE'* DATA. 469

is alfo given ; and therefore the reafon /of thief $4./>ro/.

compound line BCE to BE is given. Where-

fore by compounding, g the reason of the faidg 6.pof.

compound line BCE a:d EB, that is to fay, •

twice BC to BE is alfo given } therefore the

reafon of the only line BC to BE is given.

But the reafon of the fame BE to AB is alfo

given: Therefore/; the reafon of AB to BC ish8.$rop.

given. And feeing that the reafon of BC to

Se is given, and that as the faid BC is to BE,

fo the lquare of BC i to the reftangle of BCi 1. &

and BE, the reafon of the f^uare ot BC to the

rettangle of BC and BE is alfo given. But

the faid reftangle of EC and BE h given, it

being that which was taken away, and which k i.frof.

was given. Therefore the fquare of BC k is

given, and therefore the line BC is given.

But the reafon of the fame BC to BA is given,

therefore AB is alfo given.

PROP. LXXXVUL' , .

• ' If in a circle
ABC, given by mag*
vitnde , there be
drawn a right line
AC, which Jballtake
away a fegmevt
ABC, which doth
comprehend a given
angle AEC, that
line AC is given by
. magnitude.
Conftr. For let D be the center of the circle ;
and let the diameter thereof ADE be drawn,
and let EC be joined.

Demonp. Forafmuch as the angle ACE is
given, for a it is a right angle. But the angle
AEC is alfo given, and therefore the other
angle CAE is given. Wherefore the triangle

G g 3 ACE

a ;i l

Digitized by

47o EUCLIDE'j DATA.

b 4Q.fiyp. ACE b is given by kind i and therefore tha
ieafon of EA to AC is given. • But AE is
yen by magnitude, feeing that the circle AdC

C t.fwp. is given by altitude. Therefore c AC is
*lfo given by *nagnitudc.

PRO P. LXXXIX.

•

If inn circle ABC, given by magnitude, there
h drawn a rjgbt line AC, given by magnitude, that
line AC viU take May a fegment ABC, compre-
hending a given angle.

Conjtr. For having taken the point D for the
center of the circle, Jet the diameter ADE be
drawn, as alfp the right line EC

Demonftr. ForafamcJi as each of the right
lines AE and AC are given, the rcafon of the
a i.prop* line AE to AC a is given $ and the angle ACS
b 4l*frof.is a right angle : Therefore b the triangle ACE

is given by kand, and therefore the angle AEC e
|s given.

* PROP. XC.

If in the circumference of a circle ABC given
by pofition, and by magnitude, there be taken a
given point B9 and that from that point B, to the
circumference of the circle ABC, time doth bend
4 right line BAC, making a given angle BAC
the othfr extremity C of the bent line Jbatl be
given, '

Conflr. For let the center of the circle be D,
apd let the ri^ht lines BD and BC be drawn.

Vmonjtr.

Digitized by Google

EtfCUDE', DATA, 47I

Demon/lr. forafmuch as
each roint B and D is gi-
ven, the right line BD, ** ^.p0f%
xs given by pofition ; and
feeing that the angle BAG
is given, the angle BDC
is alfo given. ty herefore
to the right line BD gi-
ven by pofition, and in
the point D given there-
in, there is drawn the
right line CD; which
makes the given angle BDC, and therefore bb ip.prof.
the line DC is given by pofition. But the cir-
cle ABC is given by poiuion and magnitude :
Theiefore c the right line DC is given by. po-c 6.def.
fition and by magnitude. But the point I) is
given : Therefore a the point C is alto given, d zj.frof.

PROP. XCI.

If from a given
point C, there be
drawn a tight line
CJ, which fiall touch
a circle JB, given by
pofition ; that line CA
is given by pofition
and by magnitude.

Conp.Yox having
taken the point D
for the center of the circle, let the right lines
DA and DC be drawn.

Demonftr. Forafmuch as each point C and D
is given, the right line CD a is given by po- a z(9prop.
fition and by magnitude. But the angle CAD
b is a right angle ; and therefore the femicircle 5 ,gt %
defcribed on CD lhall pafs by the y oint A :
Let ij then pafs by that point, and let the
lemicircle be DAC : Forafmuch as the fame

Gg4 DAC

Digitized by Google

47* EUCLIDE'i DATA.

c6. def. DAC c is^given by pofition, and alfo the circle
d z$. prop. ABt, d the point A is given. But the point .
*e z6.frop. C is alfo given : Therefore e the Tight line
AC is given by pofition and by magnitude,

PROP. XCIL

f ^ A If without a eir-

jf >v fofition, there be

J i ] >y D> and from that

\B yC JJ given point there

\ J be drawn a right

^ y line DB , cuttivg

the circle* the reft-
^ angle comprijed wn-

<fey /fe wiofe /iw* 2?D, and the part DC, between
the point D, d«i /A* circumference convex AC
JiiaU he given.

Conftr. For from the point D let the right
line DA be drawn, which fhall touch the cir-
cle in the point A.
acr.;rs/? Lemovfir. Therefore DA a is given by pofi-
tion and magnitude \ and theietore the fquate
VS* -prof. 0f the faid T) A is b given. But the laid
c I6* 3- fquare of DA is equal c to the leftangle of fcD
and DC : Therefore the faid redtangle ot BD
and DC is alio given,

■

OTHER-

•

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EUCLIDE'/ DATA.

0

OTHERWISE.

473

Conftr. Let E
be the center of
the circle, and
by the fame
center let there
be drawn from
the point D the
right line DA.

Demotiftr.Fot-
afmuch as each
point D and £
is given, the right ]ine DE is d giveu by po- d z6.p^op.
fit ion and by magnitude. But the dicle ABC
is given by pofition and by magnitude : There-
fore each point A and F e is given, and the

Joint D is aifo given ; and therefore d each* z$*prop.
ine AD and FD is given. Wherefore the re&-
angle of the lines AD and DF is alfo given.
But the faid reftangle of AD and DF is equal
to the rectangle or DB and DJ : Therefore
the reftangle 01 DB and DC is given.

PRO P. XCIII.
m

If in a circle gi-
ven hy fofition there
he taken a given
point Ay and by that
point J there he
drawn a right line
BC to the cuctCy the
re&angle compnfed
under the fegments of
the fame line BCJball
he given.
Conjlr. For let D
be taken for the center of the ciicle, and

having

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474 1 EtJCLIDE's PJT4.

having drawn the right line AD prolong it
to the poipg E «d F. ,
Demonfir. Forafmuch as each point A and

portion. But the circle BEC is alio given by
pofition : Therefore each point £ and F is
alfo given by pofition, and the point A is gi-
bi<.2. ven. Whetrfore each line I AE and AF is
given : Therefore the reftangie of the fame
lines AE and AF is given ; and is equal to
the reftangie b of AB and AC : Therefore the
Uii re&angle of AB and AC is given.

prop. xenr.

// in a ciuh ABC, gi-
ve* hp magnitude, these be
drawn a right line B C,
which doth take away a
fegment which doth com-
prehend a given angle
JBCy mi that tie fa id
angle being in the feg-
went is cut into two equal
farts, the line compounded
of the right lines MA and
AC, which comprehend the gmen angle BAC JbalL
have A given tea/on to the line AD, which
4oth divide that angle into two equal parts ; and
the re&angle contained under the line compounded
<>f thofe lines MA and AC, comprehending thegi-
vai angk BAC, and that part ED of the interfer-
ing line which is below the fegment between the
Jfafe BC and the circumference, Jball be given.
Confer. Let BD be drawn.
Dmonfiu Forafmuch as in the circle ABC
given by magnitude, there is drawn the right
hue BC, which tajces away the fegment BAC
co^prehendiflg the given angle BAC, that

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EUCUDE'i DATA. v 47*

line BC & is given ; and therefore BD is alfo a 88. fm

given : Therefore the reafon of BC to BD b is b 1. p. v.

given. And teeing that the given angle BAG

is cut in two equal \ arts by the right line AD,as

c BA is to CAt lo is BE 10 CE s and by com c 3. j

pounding, as BAG is to CA, fo is BC to Ct i

and by permutation, as BAG is to BC, fo i*

CA to CE. And feeing that the ange t A£

is equal to the angle CAE, and the a -gle

ACE d to the angle BDE, the other an^.e d zr. jv

AEC is equal to the othe< angle ABD ; and

therefore the triangle ACE is equiangled

to the triangle ABD : Therefore e as AC is to e 4. 6*.

CE, fo is AD to BD. But as AC is to CE,

fo the line compounded of BA and AC is to

BC : Therefore as the compound line BAC is

to BC, fo is AD to BD , and by permutation,

as the compound line BAC is to AD, fo is

BC to BD. But the reafon of BC to BD is

given : Therefore the reafon of the compound

line BAC to AD is alfo given. Moreover,

I fay that the rectangle under the compound

line BAG and ED is given- For feeing that

the triangle AEC is equiangled to the trian*

gle BDE, (for {he angle ACE d is equal to the

angle BDE, and the angle AEC/ to the angle* XJ*

BED) as BD is to DE, fo is AC to CE. 0ut

as AC is to CE, fo is alfo the compound

line BAC to BC : Therefore as the compound

line BAC is to BC, fo is BD to DE. Where*'

fore the rectangle of the compound line BAC

and DE g is equal to the rectangle of BC and g 16*. 6.

BD. But the redangle of BC and BD is

given, (for that thofe lines BC and BD ace

given :) Therefore the reftangle under the com*.

pound line BAC EP is alio give*

OTHER-

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47^

EUCUDE'* DATA.

OTHERWISE.

h 32.1
i 5. u

Conftr. LetCA
be prolonged to
the point E, and
let AE be put
equal to BA,
and let BE and
BD be joined.

Dcmonjlr. For-
afmuch as the
angle BAC is
double to each
or the angles
CAD and AEB
(for the angle
BAC i : cut in-
to two eciual
parts by the line
AD, and e^ual h

to the two angles ABE and AEB, which 2 are
equal) the angle ABE is equal to the angle
k 2i, 3. CAD, that is to fay, k to the angle CBD , ad-
ding therefore the common angle ABC, .the
whole angle ABD fliall be equal to the whole
angle FBE. But the angle ACB is k equal to
the angle ADB : Therefore the third angle
AEB is equal to the thid angle BAD; and
therefore the triangle CEB is equiangled to
the triangle ABD : Wherefore as Ch is to
CB, fo is AD to BD. But the right line CE is
compounded of the two lines CA and AB :
Therefoie as the compound line BAC is to CB,
to is AD to BD ; and by permutation, as the
compound line BAC is to AD, fo is CB to
BD. But the reafon of CB to BD is given,
feeing that each of thofe lines is given:
Therefore the reafon of the compound • linak
BAC to AD is alfo given. And feeing that?

the

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477

EUCLIDE'* DATA.

the triangle CEB is equiangled to the triangle
FBD (for the angle AFC is equal / to the an- 1 zr.
gle BFD, and the angle ECB m to the angle m 16. &
ADB) as EC is to CB, fo is BD to DF. But
EC is equal to the compound line BAC :
Therefore as the compound line BAC is to
CB, fo is BD to DF. Wherefoie n the redt-n 16.6.
angle of the compound line BAC and DF is
equal to the rectangle of CB and BD. But
the redangle of CB and BD is given, confi-
dering that each of the lines CB and BD is
given : Therefore the re£langle of the com-
pound line BAC and DF is given,

OTHERWISE.

Conftr. Let AC be prolonged to F, and let
CF be put equal to AB, and let the righ^
lines BD and DF.be drawn.

Demonftr* Forafmuch as BA is equal to CF9
and 0 BD to DC, the two fides AB and BD<>
are equal to the two fides CD and DF, each
to his correfponding fide, and the angle ABD
\s equal to the angle DCF, f feeing (hat the p xu j.

four £

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feUCLIDE/ tAfJL

.fewfided figure ABDC is within the circle*
Therefore the bafe AD is q equal to the bafe
DF, and the angle DAB to the angle DFC.
But the angle BAD is given, (being the half
of the given angle BAG ) Therefore the an-
gle DFC is fo alfo. But DAF is alfo given :
Therefore the triangle ADF is given by kind*
Wherefore the reafon of FA to AD is given.
But AF is the compound of BA and AC, for
that CF is equal to AB : Therefore the rea-
fon of the compound line BAG to AD is gi-
ven : The fame Demonftration will ferve to
fhew that the reftangle contained under the
compound line BAC and ED is given alfo.

PROP. XCV.

If in the
diameter EC
of a circle
ABC given
hy f option %
- there he taken
a given point
D, and that
from that
point D there
he drawn a
right line
t>J% to the
circumference

tfthe eirtk. But from the fe&ion of the fetid line
there he drawn a right Irne AM, Perpendicular
thereto, and by the point E where that perpendi-
cular doth meet with the circumference x there he
drawn a parallel EP, to tbefcrfi line drawn JD9
that foint F in which the parallel meets with
the diameter % is given 5 ana the reft angle con-
tained under the paraUtl Imts JD xtnd EP it
*lft> given*

Conftn

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EUGLIDE'/ DATA. 47*

Conjlr. Let the right line EF be prolonged
to the point G, and let the right line AG be
drawn.

Demonftr. Forafmuch as the angle AEG is
a right angle, the right line AG is the dia-
meter of the circle. But BC is alfo the dia-
meter : Therefore the point H is the center
of the circle. Now the point D is given ;
and therefore a the line DH is given by mag. a z6.$wf.
nitude. But feeing that AD is parallel to EG,
and AH equal to GH; b DH is equal to ¥HKb z6.pof.
and AD to FG ; (for the angles AHD and
FHG c are equal, aud DAH and FGH d arec 15. r,
alfo equal.) But the line DH is given :d 19.1.
Therefore FH is alfo given. But each of thofe
lines DH and HF is alfo given by portion,
and the point H is given : Therefore e the e zj.frof.
point F is alfo given. And feeing that in
the circle ABC given by pofuion, is taken
the given point F, and through the fame is
drawn the right line EFG ; the reftangte un-
der EF and FG / is given. But FG is equal f 9J. *«fi
to AD. Therefore the reftangle comprehend-^
ed under AD and EF is given. Which was t*
be demonftrated.

Ihc End of EUCLIDEV DM

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■» « ♦
■

»

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« . »

w i r

. ... J

4« «r

• ■

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» » « . *

BRIEF TREATISE,

Added by FlUSSAS,

«• • • # • «

Urn 9 # •* »• »• ♦ •

Regular Solids.

Egular Solids arc faid to be com*
pofed and mix'd when each of
them is transformed into other
Solids, keeping ftill the form,
number and inclination of the
bafes, whiph they before had to one ano-
ther } fome of which yet are transformed into
mix'd Solids, and other fome into fimple.

H h Into

48* A TREATISE of

Into mixt, as a Dodecaedron and an Icofa-
edron, which are transformed or altered, if
you divide their fides into two equal parts,
and take away the folid angles fubtended of
plain fuperficial figures, made by the lines
coupling thofe middle feftjons ; for the So*
lid remaining after the taking away of thofe
folid angles, is called an Icofidodecaedron.
If you divide the fides of a Cube and of an
Odloedron into two equal parts, and couple
the feftions, the folid angles fubtended of
the plain fuperficies made by the coupling
lines, being taken away, there (ball be left a
folid, which is .called an Exoftoedron. So

5 hat both of a Dodecaedron and alfo of an
cofaedron, the Solid which is made ihall be
called aq lcofidodecaedron; and likewife the
Solid" made of a Cube, and alio of an Ofto-
edron, ihall be called an Exoftoedron. Buj the
other Solid, to wit, a Pyragiis or Tetraedron,
is transformed into a fimple Solid $ for if you
divide into two equal parts each of the tides
of the tyraipis, * triangles defcribed of the
liqes wjnch couple the feftions, and fubtend-
ing and taking r?way the folid angles of the
Pyramis, are equal and like unto the equi-
lateral triangles left in each of the bafes, of
all which triangles is produced an O&oedron,
to wit, a fimple* and not a compofed So*
lid* For the OAoe^rop ha>th four bafes, like
in number, form, and mutual inclination with,
the bales of the pyrapiis, and hath the other
four bafes with like fituation qppofite and pa-
Tallel to the Jormer. Wherefore the applica-
tion of the pyramis takeji twice^ maketh £
fimpie Oftoedron, as the othqf Solids majcp
a mixM compound Solid,

DEFI-

REGULAR SOLIDS.
. DEFINITIONS.

I. An Exottoedron is a [did figure con-
tained of fix equal fquares, and eight
equilateral and equal triangles*

II. An Uolidodec^edron is a fo&4 ' figure
contained under twelve equilateral, zquai,
and tquiangkd Pentagon*, and twenty
€<}ual ajtd trilateral triange^ * "

For the betttr Underftanding of the two
former Definitions, and alfo or the two Pro-
portions following-, I have here fet two fi-
gures, whofe figures if you firft defcribe upon
pafted Paper or fuch like matter, and then
cut \them and fold them accordingly, they
will reprefent unto you the perfeft forms of an
Exoftoedron, and of an Icofidodecaedron.

a

PR0BLEME

To defcribe an equilateral and) equiangled
Extftoedron, and to contain it - in a given
Sphere, and to prove that the Diameter of
t he Sphere is double to the fide of the /aid
Exo&oedron.

Conftr, Suppofe a Sphere whofe diameter let
be AB, and about the diameter AB let there
a 6. 4. be defcribed a fquare a , and upon the fquare
b 15. 15. let there be defcribed a Cube b, which let be
} CDEFQTVR ; and let the diameter thereof
be QR, and the center S. Divide the fides
of the Cube into two equal parts in the
points G, H, I, K, L, M, N, O, P, &e. and
couple the middle fedtions by the right lines
IN, NO, OP, PI, and fuch like, which fub-
tend the angles of the fquares or bafes of the
c 4. I. Cube j and they are equal c, and contain right
angtes, as the angle NIP. For the angle
which is at the bafe of the Ifofceles

\ triangle

angle

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REGULAR SOLIDS.

triangle NDI, ii the half of a right angle, arid
fo liferwife is the oppolite angle RIP. Whefc-

fore the refidue NIP is a right angle, and fo
the reft. Wherefore NIPO is a fquare. And
by the fame reafon fhall the reft NMLK,
KGHI, &c. inferibed in the bafes of; the
Cube, be iquares, and they fhall be fix in num-
ber, according to the number of the bafes
of the Cube. Again, forafmuch *as the tri-
angle KIN fubtendeth the folid angle D, of
the Cube, and likewife the triangle KGL the
folid angle C, and fo the reft which iubtend
the righc folid angles of the Cube, arid thefe
triangles are equal and equilateral (to wit) be-
ing made of equal fides, and they are the li-
mits or borders of the fquares, and the ftmates
the limits or borders of them \ as hath been

H h | brio**

486 2 TREATISE of

before proved. Wherefore LMNOPHGK is an
Exoftoedron by the definition, and is equi-
lateral ; for it is contained of equal fubtendant
lines, it is alio equiangled ; for ever? folid
angle thereof is contained under two fuperfi-
cial angles of two fquares, and two fuperficial
angles of two equilateral triangles.

■ vemonftt. Forafmuch as the oppofite fides
an d diameters of the bafes of the Cube are

{tafallels, the plain extended bf the right
tnes QT and VR, lhall be a parallelogram.
And for that alfo in that plain lyeth QR, the
diameter of the Cube, and in the fame plain
alfo is the line MH, which divideth the faid
plain into two equal parts, and alfo coupleth
*he oppofite angles of the Exoftoedron j this

line MH therefofe divideth the diameter into

two

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REGULAR SOLIDS. 4°7

two equal parts i j and alio divideth it fclfd cot.^v
in the fame point, which let be S, into two
equal parts e. And by the fame reafon maye4,,,
wV move that the reft of the lines which
couple the oppohte angles of the Ejpcloedton,
do in S the center of the Cube, divide one
another into two equal parts, for each or the
ancles of the Exoftoedron ate fet in each ot
the8 oafes of the Cube. . Wherefore making
the center the point S, with the
or SM defcribe a Sphere, and it Mwuch
every one of the angles equidiftant from the

^And\rafmuch as AB the dUmeter of i the
fphere given, is put equal to diameter of
the bafe of the cube, to wit, to the une
RT, and the fame line RT is equal to the line
MH f, which line MH coupling the oppofitef }J. I.
angles of the Exoftoedron, is drawn by the
center. Wherefore it is the diameter of the
Sphere given which containeth the Exotto-

' Laftly> forafinuch as in the triangle RFT,
the line PO doth cut the fides into , two

SSf RT £ft ™T B* FR R^°obr1Ctheg
FP by luppofition: Wherejiwe RJ. 01
uiamier HM, is alfo double to tto ljg PO.
the fide of the Exoftoedron. Wherefore we
have defcribed, &c. Which was required to
be done.

PROBLEMS II.

To defcribe an trilateral and tautan-
gled Iccfidodtcaedron, and to cotopebtnd
it in a ftkm given, and to frovt that tbt

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488 U TREAflSE of

diameter being divided by . an exttemt

and mean ftpfo^UH^ mahtb tbt greater

fegment double to the fide of the Icofi-

dodecaedron. • ; . »

Gonftr, Suppofe that the diameter of the

t jo. 6. fphcre given be NL> a divide the ljine NL|
ly an extreme tod mean proportion in the
point I, told the greater feginent thereof let
be NI? And upon the line NI 4ef$ribe a

b 15, 15. Cube * 5 and ibout tliis C^beikt the" be cir-

c 17. 12. cumfcribed a Dodecaedron c ; and Jet the
farte be ABCDEFHKMO, arfddiv^ each of
the fide* into two etual parti in (he ppints
Q, R, i5, T, V, Xv Yf Z, P? e, *&, Qt
. Md couple the feftaofU with right Uses, which
fhall fubtend the angles of die Pentagons, as
the lines PQj OVf VQ,QX, YR, RQ, V?,
TI, XV, and to the reft,

Demonfir. Foraforuch as thefe lines fubtend
equal angles of the Pentagons, and thofe equal
. angles are contained of equal fides, ta wit, of
the halfs df the fides of the . Pentagons ;

<1 4. I. therefore tb6fe fubcending lines are equal /.

Wherefore tte triangles GQV, YQfc, and
VXT, and* the reft, which take away folid
ancles of the Dodecaedron, tteiequilnterah

Ag;ain, fotafehuth .As m evely Pentagon are
defer ibed fi^e equal night linte, Oouphng the
middle fe&ltiH9>fc£ the fides, M Are the lines
* QV, VT, TS, SR, and RC[, they defcribe a
Pentagon in the plain of the Pentagon of
the Dodecaejlroj.^ Aad the f aid - Pentagon is
contained in a circle, to wit," whofe center
is the center x>L a Pentagpn of the . Dotkca-
edron. For the lines drawft * frorti that center
to the angles of this Pentagon are ecfual, for
that they art perpendiculars upon the bates
■ •* * ! . cut.

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REGULAR SOLtDS. 48*

cut r. Wherefore the Pentagon QJEtSTV, h e ii. 4.
equiangled/. And by the latae ieafort mayf n. 4.

— *

* fr- "

the reft of the Pentagons defcribtd In the
bafes of the Dodecaedron^ be pored equal and

like. flv

Wherefore thofe Pentagons are twelve in
number : And forafmuch as the equal and lite
triangles do fubtend and take away twtntf fo-

lid angles of the Dodecaedron ; therefore the
laid triangles fhall be twenty in number.
Wherefore we have defcribed an Icoiidodeca-
edion by the Definition, which Icofidodeca-
edt$n is equilateral $ for that all the fides of
the triaTftgles are equal and common with the
Pentagons \ and it is alfo equiangled. F01
each of the folid angles is made of two fu-

perficial

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490 A TREATISE of

perficial angles of an equilateral Pentagon,
and of two fuperficial angles of an equilate-
ral triangle.

Now let us prove that it is contained in
the given fphere whofe diameter is NL. For-
afmuch as perpendiculars drawn from the cen-
ters of the Dodecaedron, to the middle fe-
ftions of his fides, are the halfs of the lines
which couple the oppolite middle feftions of
g i tor. of the fides of the Dodecaedron g $ which lines
17- 1 j. alfo h do in the center divide one another int0
b idcnu two equal parts. Therefore right lines drawn
from that point to the angles of the Icofi-
dodecaedron (which are fet in thofe middle
feftions) are equal ; which lines are thirty irt
number, according to the number of the fides
of the Dodecaedron i tot each of the angles
of the Icofidodecaedroti are fet in the middle
feftions of each of the fides of the Dodeca*
edron. Wherefore making the center of the
Dodecaedron, aqd the fpace any one of the
lines drawn from the center to the middle
feftions, defcribe a fphere, and it ihall pafs by
all the angles of the Icofidodecaedron, and
lhall contain it.

And forafmuch as the diameter of this fo-
lid, is that right line whofe greater feg-
ment is the fide of the Cube infcribed in
1 4* cor. Dodecaedron i, which fide is NI by
fuii. ' fuppofition. Wherefore that folid is contain-
ed in the fuhere given, whofe diameter is put
to be the line NL.

Now

■

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Now let us prove that the greater fegment
of the diameter is double to QV the Sde of
the folid. Forafmuch as the fides of the tri-
angle AEB, are in the points Q and V divi-
ded into two equal parts, the lines QV and
BE are parallels k. Wherefore as AE is to
A V? fo is EB to VO /, But the line AE is 1 *• 6.
double to the line AV. Wherefore the line
BE is double to the line Qy». Now the 014*6.
line BE is equal to NI, or to the fide of the
Cube 11 ; which line NI is the greater fegment n a w.of
©f the diameter NL. Wherefore the greater 17* lh
fegment of the diameter given is double to the
fide of the IcofidodecaedroQ inferibed in the
given fphere. Wherefore, We luve defcri-
bed, fife, Which was required to W done.

JDFZX^

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3 treatise ,f

* *

. * ADVERTISEMENT.

To the IJnderftanding of the nature bf this
Icofidodecaedron, you muft well conceive the
paffions and proprieties of both thefe fdlrds, of
whofebafes it confifteth, to wit, ofthelcola-
cdron and of the Dodecaedron. And altho' in
it the bafes are placed oppofitely, yet have
they to one another one and the fame incli-
nation. By reafon whereof there lye hidden in
it the aftions and paffions of the other Regu-
lar Solids. And I would have thought it not
impertinent to the purpofe to have fet forth
the infcriptions and cxrcumfcriptions of this
Solid, if want of time had not hindred. But
to the end the Reader may the better attain
to the Underftanding thereof, I have here fol-
lowing briefly fet forth, how it may in or
about every one of the five Regular Solids be
infcribed or circumfcribed ; by the help where-
of he may, with fmall travel or rather none at
all, having well poifed and confidered the De-
wonftrations appertaining to the forefaid five
Regular Solids, demonftrate both tilt inscri-
ption of the faid Solids in it, and the irtfcri-
ption of u in the faid Solids.

#

Of the Infcriptions and Cwcumferipions of

an Icofiiobc&cdron.
»«.. - * » » «

An Icofidodecaedron may contain the other
five regular bodies. For it will receive the
angles of a Dodecaedron in the centers of the
triangles which fubtend the folid angles of the
Dodecaedron, which folid angles are twenty
in number* and are placed in the faine order
in which the folid angles of the Dodecaedron
taken away, or fubtended by them, are. And
by that reafon it ftall receive a Cube and a

Pyramis

i

s

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REGULAR SOLJDS.

Pyramis contained in the Dodecaedron, when
as the »ng^ of the one, a±e fee in the angles
of the other. ^

An Icoiidodecaedron receiveth an Oftoedron,
mxht angles cutting the fix oppofue lections

of the pqdecacdron, even as if it were a fimple

Qodecaedron- lit w : u %iiiff \ *l c

And it contained) an Icofaedron, placing the

twelve angles of the Icofoedron in the Tame
centers of the twelve Pentagons. ,
It may alfo'by the fame reafon be infer**;
bed in each of the five regular bodies, to wit,
ijV a Pyramis, if you place four triangular ba-
fes concerurical with four bafes of the Pyramis,
after the fame manner that you inferibed an
Icofaedron in a Pyramis 5 fo hkewife may at
be iufcrjbed in an O&oedron, if you make eight
bafes thereof concentri^al with the eight bafes
of the O:loedron. It fhall alfo be inferibed
in a Cube, if you place the angles which re-
ceive the O&oedron in it, in the centers of
the bales of the Cube. Again, you {hall in-
Ccribe it in an Icofaedron, when the triangles
cgmpaffed in of the Pentagon bafes, are can*
centrical with the triangles which make a
folid angle of the Icofaedron. . mm j

kaflly, It fhall be inferibed in a Dodeca*
qdrw, if you place &ch of the angles there^
of in the middle fedions of the fides of the
Dodecaedron, according to the prd^r of the
Confirmation thereof. u.)

The oppotite plain fuperlicies alfo of this
folid are parallels. For the oppofue, folid an-
gles are fubtended of parallel plain fuperlicies,
as well in the angles of the Dodecaedron fub-
tended by triangles, as in the angles of the
Icofaedron fubtended of Pentagons, which
thing may eafily be demonftrated. Moreover,
in this folid are infinite properties and pafiions,
fpr^oging of the foli4s whereof it is cwnjoM.

494 r^ TREATISE of

Wherefore it is manifeft, that a Dodeca-
edron and an Icofaedron mixed, are transform-
ed into one and the felf fame folid of an lcm-
fidodecaedron. A Cube alfo and an Oftoedron
are mixed and altered into another folid, to
tirit, into one and the fame Exo&oedron. But
a Pyramis is transformed into a fimple and
perteft folid, to wit, into an OAoedron.

If We will frame thefe two folids joined to-
gether into one folid, this only muft we ob- *
ferve.

In the Pentagon of a Dodecaedron infcribe
a like Pentagon, and let its angles be fet in
the middle leflions of the Pentagon circum-
scribed, and then upon the faid Pentagon in*
fcribed, let there be fet a folid angle of an
Icofaedron, and fo obferve the fame order in
each of the bafes of the Dodecaedron, and
the folid angles of the Icofaedron fet upon
thefe Pentagons fhall poduce a folid conhft-
ing of the whole Dodecaedron, and whole
Icofaedron. In like fort, if in every bafe of
the Icofaedron, the fides being divided into
two equal parts, be infcribed an equilateral
triangle, and upon each of thofe equilateral
triangles be let a folid angle of a Dodecaedron,
there (hall be produced the fame folid con lift-
ing of the whole Icofaedron, and of the whole
Dodecaedron.

And after the fame order, if in the ba»
fes of a Cube be infcribed fquares fubtending
the folid angles of an Octoedron, or in the
bales of an O&oedron be infcribed equilateral
triangles fubtending the folid angles oi a Cube,
there lhall be produced a folid confifting of
either of the whole folids, to wit, ©t the
whole cube, and of the whole Oftoedron.

But equilateral triangles infcribed in the
bafes of a Pyramis, having their angles fef in
the middle feclions of the iides of the Pyramis,
' ' an4

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REGULAR SOLID&

and the folid angles of a Pyramis, let upon
the faid equilateial triangles, there fhall be
produced a folid confifting of two equal and
like pyramids.

And now if in thefe folids thus compofed,
you take away the folid angles, there fhall be
reftored again the firft compofed folids, to
wit, the folid angles taken away from a Do-
decaedron and an Icofaedron compofed into
one, there fhall be left an Icofidodecaedron,
the folid angles taken away from a Cube and
am O&oedron compofed into one folid, there
fhall be left an Exo&oedron. Moreover, the
fplid angles taken away from two pyramids
compofed into one folid, there fhall be left
an Oftoedron.

Of tie nature of a trilateral and equi-
lateral Fjramis.

♦

i. A trilateral equilateral Pyramis is divi-
ded into two equal parts, by three equal
fquares, which in the center of the Pyramis
cut one another into two equal parts, and
perpendicularly, and whofe angles are let in
the middle feftions of the fides of the Py-
ramis.

a. From a Pyramis are taken away four
Pyramids like unto the whole, which utterly
take away the fides pf the Pyramis, and that
which is left is an Oftoedron inscribed in
the Pyramis, in which all the folids infcribed
ip the Pyramis are contained.

3. A perpendicular drawn from the angle
pf the Pyraqiis to the bafe, is double to the
diameter of the Cube infcribed in it.

4. And a right line coupling the middle
feftions of the oppofite fides ot the Pyramis
is triple to the fide of the fame Cube.

' The

4 TREATISE of

5. The fide alfo of a Pyraijris is triple to tfre
diameter of the hafe of jhe Cube* t

6. Wherefore the fame fide of the Pyramis
is in power double to the right line which
coupleth the middle feftions of the oppofite

7. And it is in power fefquialter to the per-
pendicular which is dmwn from the angle to

the bafe. ^

8. Wherefore' the perpendicular is in power
fefquitertia to the line which coupleth the
middle fedions of the oppofite fides.

9. A Pyramis and an O&oedron inferibed in
itf alfo an Icofaedren inferibed in the fame
OAoedron , do contain one and the fame
fphere.

■ „ • . *

Of tit nature \cf an Oftotdrvn*

* • * *

T. Four perpendiculars of an O&oedron,
drawn in four bafes thereof from two 'oppo-
fite angles of the faid O&oedron, and cou- v

Sled together by thofe four bafes, defciibe a
Lhombus, or Diamond figure 5 one of whofe
diameters is in power double to the other
diameter.

2. For it hath the fame proportion that the
diameter of the Oftoedron hath to the fide of
the (Moedron. " A

$. An O&oedron and an Icofaedron in-
feribed in it, do contain one and the fame
fphere. •* * *

4. The diameter of the folid of the O&o~
cdron is in power fefquialter to the diame-
ter of the circle which containeth the bafe,
and is in power duple fuperbipartiens tertias
(that is, as 8 to j,) to the perpendicular or
fide of the forefaid Rhombus ; and moreover

* » • 2 r»

V

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REGULAR SOLIDS.

is in length triple to the line which coupleth
the centers of the next bafes.

j. The angle of the inclination of the bafes
of the Oftoedton, doth witb the angle of the
inclination of the bafes of the Pyramis, make
angles equal to two right angles.

>

Of the natute of a Cubel

t . The diameter of a Cube is in power fe't
quialter to the diameter of his bate.

i. And is iri power triple to his fide.

j. And unto the line- which coupleth the
centers of the next bafes, it is in powec
fextuple.

4* Again, the fide of the Cube, is to the
fide of the Icofaedron inscribed in it, as the
whole is to the greater fegment.

Unto the fide of the Dodecaedron, it is
as the whole is to the leffer fegment.

6. Unto the fide of the Oftoedron it is in
power duple.

7. Unto the fide df the Pyramis it is iit
power fubduple.

8. Again, the Cube is triple to the Pyra-
mis, but to the cube the Dodecaedron is in
a manner double. Wherefore the fame Dode-
caedron is in a manner fextuple to the laid
Pyramis*

Of the nature of the Icofaedron.

t. Five triangles of an Icofaedron, do make
afolid angle, the bafes of which triangles make
a Pentagon. If therefore from theoppofite ba-
fes of the Icofaedron be taken the other Pen-
tagon by them defcribed, thefe Pentagons lhall
in fuch fort cut the diameter of the Icofaedron
which coupleth the forefaid oppofite angles,

I i that

■

A TREATISE tf

that that part which is contained between the
planes of thefe two Pentagons fhall bfe. the
greater fegment, and the refidue, whicft is
drawn from the plain to the angle, fhall be
the leffer fegment.

z. If the oppofite angles of two bafes join- '
ed together, be coupled by a right line, the
greater fegment of that right line is the fide
of the Icofaedron.

5. A line drawn from the center of the Ico-
faedron to the angles, is in power quintuple
to half that line, which is taken between the
Pentagons, or of the half of that line, which
is drawn from the center of the circle which
containeth the forefaid Pentagon, which two
lines are therefore equal.

4* The fide of the Icofaedron containeth in
power either of them, and alfo the leffer feg-
ment, to wit, the line which falieth from the
folid angle to the Pentagon.

5. The diameter of the Icofaedron containeth
in power the whole line, which coupleth the
oppofite angles of the bafes joined together/
and the greater fegment thereof, to wit, the
fide of the Icofaedron.

6. The diameter alfp is in power quintuple
to the line which was taken between the Pen-
tagons, or to the line which is drawn from the
center to the circumference of the circle which
containeth the Pentagon compofed of the fidds
of the Icofaedron.

^ 7. The dimetient containeth in power the
tight line which coupleth the centers of the
' oppofite bafes of the Icofaedron, and the dia-*
meter of the circle which containeth the bafe.

8. Again, the faid dimetient containeth in
power the diameter of the circle which con-
taineth the Pentagon, and alfo the line which
is drawn from the center of the fame circle
*® the circumference j that is, it is quintuple

4

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REGtfLAR SOLIDS.

tb the line drawn from the center to the cir-
cumference.

9. The line which coupleth the cehters of
the cppofite bafes, containeth in power the
line which coupleth the centers of the next
bafes, and alfo the reft of that line of which
the fide of the Cube inferibed in the Icofaedron
is the greater fegment.

to. The line which coupleth the middle le-
gions of the oppofite fides, is triple to the
fide of the Dodecaedron inferibed in it,

ix. Wherefore if the fide of the Icofaedron,
and the greater fegment thereof be made one
line, the third part of the whole is the fide of
the Dodecaedron inferibed in the Icofaedron.

Of the Dodecaedron.

1. The diarheter of a Dodecaedron contaiheth
in ppwer the fide of the Dodecaedron, and al-
fo that right line to which the fide of the
Dodecaedron is the letter fegment, and the fide
of the Cube inferibed in it is the greater feg-
rnent, which line Is that which fubtendeth the
angle of the inclination of the bafes, contained
under two perpendiculars of the bafes of the
Dodecaedron;

2. If there be taken two bafes of the Dode-
taedron, diftantfrom one another by the length
of one of the fides, a right line coupling their
centers being divided by an extreme and mean
proportion, maketh the greater fegment the
right line which coupleth the centers of the
next bafes.

j. If by the centers of five bafes fet upon
one bafe, be drawn a plain fuperficies, and by
the centers of the bales which are fet upon
the oppbfite bafe, be drawn alfo a plain fu-
perficies, and then be drawn a right line,
coupling the centers of the oppofite bafes,

I i z that

rA TREATISE of

that right line is fo cut, that each of his
pans let without the plain fuperficies, is the
greater fegment of that part which is con-
tained between the plains*

4. The fide of the Dodecaedron is the greater
fegment of the line which fubtendeth the an-
gle of the Pentagon.

$. A perpendicular line drawn from the cen-
ter of the Dodecaedron to one of the bafes,
is in power quintuple to half the line which
is between the plains. •

6. And therefore the whole line which cou-
pleth the centers of the oppofite bafes, is in
power quintuple to the whole line which is
Between the laid plains.

7. The line which fubtendeth the angle of
the bafe of the Dodeoaedron, together with
the fide of the bafe are in power quin-
tuple to the line which is drawn from the
center of the circle which containeth the bafe,
to the circumference.

8. A fe&ion of a fphere containing three
bafes of the Dodecaedron, taketh a third part
of the diameter of the laid fphere.

9. The fide of the Dodecaedron and the line
which fubtendeth the angle of the Pentagon,
are equal to the right line which coupleth
the middle fe&igns of the oppofite fides of the
Dgdecaedrgn,

THE

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— 1 "

THE

THEOREMS

O F

»

ARCHIMEDES.

Concerning the Sphere and Cylin-
der, Inveftigated by the Method
of Indiviftbles, and briefly De-
monftrated by the Reverend and
Learned Dr. Ifaac Barrow.

TH E main Dcfign of Archimedes in
his Treatife of the Sphere and Cy-
linder, is to refolvc thefe four Pro-
blems.

t. To find the proportion of the fuperficies of a
fkbere to any determinate circle ^ or to find a cir-
cle equal to the fuperficies of a given fihere.

2,. To find the proportion of. the fuperficies of
a%y fegvient of a jfibere to any determined circle i
ov to find a circle cyual to the fuperficies of any
tfrfi fegment. |f> ^

3. To find tit proportion of tie Jphere it J elf
(or of its folid content) to any df terminate Cone or
Cylinder ^ or to find a Cone or Cylinder equal to

' 4 given jpiere.

4. To find the proportion of a fegment of a $lerf
to any determinate Cone or Cylinder ; or to find 4
Cone or Cylinder equal to a given fegment.

Thcfe four Problems Archimedes profecutes
fcparately, and lays down Theorems imme-
diately fubfervient to their folution ^ but we
reduce them to two : For fince an Hemi-
Inhere is the fegment of a fphere, and the
method of finding out its relations, in refpeit
to the fuperficies and folid content, is com-
prehended m the general method of invefti-
gating the proportion of the fegments : And
t'rom the fuperficies and folid content of an
Hemifpheje already found, the double of them,
(that if, the fuperficies and content of the
whole fphere) is at the fajne time given. Anp!
jndeed 'tis lupeifluous and foreign from the
Laws of good Method, to inyeftigate theif
relations diftin&ly and fcparately ; To that if
it were not a crime, I might on this account,
blame even Jnhimedes himfelf.

The whole matter therefore is redue'd to
thefe 'two Problems.

' r. To find the proportion of the fuperficies of
any fegment of a Jfobert9 to a determinate circle j
of to find a circle \ equal to the fuperficies of a
given fegment.

^ z. To find the proportion of the folidity of any
fegment of a §hcre to any determinate Con* or
Cylinder y or to find a Cone or Cylinder eaual to
an ajjign'd fegment of a fphere.

I fhaU relolve thefe Problems by another
much eafier and fhorter method: In which the
order being inverted, : firft, 1 fhall ftek the
folidity of a fegpaen*, and from thence deduce
•:' . ; its

Digitized by Google

its fuperficies ; a thing which is in my judg-
ment well werth obferving, and pexform d, as
I know of, by none.

Firft therefore, for finding the folidity of a
fegment, I fhall lay down two, commonly
known and received Suppoiitions, viz.

j. That a f$ries of magnitudes proceeding in
Arithmetical Progrejfion from nothing (inclufive) or
whofe common difference is equal to the leaf mag-
liitudey is fubduple of as many quantities equal
to the greatejl ; (i. e. fubdupie of the' product of
the greatcfi term and number of terms : ) So that
if the fum of the terms be called z, the greatejl
term g% and the number of terms nt then will

z — -5 , or z z — ng .

The truth of this Proportion will eafily
appear by expreffing the leries twice, and in-
verting the order 5

•

o, a, za, 3a, 4a*
4a, 3a, za, a, o.

For fo the difference always being equal to
the leaft quantity, 'twill be evident that each
two correfpondent terms taken together are
equal to the greateft term ; and alio, that the
feries taken twice is equal to the greateft term
repeated as many times as there are terms,
i. e. the laft term drawn into the number of
terms.

We have in a triangle a very clear and ea-
fy example of this moft uletul Proportion,
which is prov'd hence, to he half a parallelo-
gram having the fame altitude, and Jlanding on
the fame bafc*

Suppofe

I 5°4 3

Suppofe tfoe altitude
AE of the triangle AEZ
to be divided into parts
indefinitely many and
fmall AB, BC, CD, DE,
and parallels BZ, CZ,
DZ, EZ, drawn thro* the
points of Divifions ^ all
thefe proceed from o in
an Arithmetical Progreffion, and confequent-
ly the fum of 'em all, that is, the triangle
AEZ, is fubduple of the greateft EZ drawn
\nto the altitude AE, by which the fum of
the terms is exprefsM, that is, fubduple of the
Parallelogram EY, whofe bafe is EZ, and
altitude AE.

But the illuftration of the Rule will con-
duce more to our defign by inferring hence,
That a circle is equal to half of the radius drawh
into the cir$umference9 after this manner* Con-
ceive a circle to confift of as many concentric
Peripheries as there are points or equal parts
indefinitely jnany and fmall in the radius,
Thefe Peripheries, as well as their radii pro-
ceed from the center or nothing in an Arithme-
tical progreffion j and therefore their fum, that
$s, the whole circle is equal to half the greateft
(or extreme circumference) drawn into the
number of terms, that if, the' radius.

After the fame manner we may fuppofe
the feftor AEZ to confift of as many con-
centric Arcs BZ, CZ,
DZ, EZ, as there are
points (or equal parts
indefinitely fmall) in the
tadius AE, which Arcs,
as their radii, proceed*
jqg f rem a point or no-
thing iirt an Arithmeti-
cal progreffion, the feftor

alfo

Digitized by Google

alfo will be equal to half the radius drawn in?
$o the extreme Arc EZ. Which may be
made evident alio after this manner : Let us
fuppofe the right line EY to be perpendicu-
lar tp the radius AE, draw the right line AYt
and from the points B, C, D, of Bivifion in
the radius, draw BY, CY, DY, parallel to EY,
and terminated at AY. Becaufe EY: DY ( :: rad.
AE : rad. AD) :: Arc EZ : Arc DZ. and EY =
EZ, then will DY = Arc DZ ; and in like
rnanner will CY=CZ, and BY = BZ. whence
the triangle AEY will be = to the feftor

v.- AE x EY ,AExEZ, r
AEZ, that is, — ( ) = fe-

X 2

ftor AEZ. By this means we colleft tfhat ce-
lebrated Theorem of [Archimedes, That a circle ise-
qual to a triangle, whofe haje is equal to the radius,
and altitude equal to the periphery of the circle ; and
that without any imcription or circumscri-
ption, of figures, by only fuppofing that the
Area or Superficies of the circle cpnfifts of in-
finitely many concentric Peripheries. Which
method of indivifibles, (now firft of all known to
me) feems no lefs evident (nay more evident)
and perhaps lefs fallacious than that wherein
planes are fuppofed' to cpnfift of parallel right
lines, and folids of parallel planes ; as hereT
after fhall be evident, when we fljall colleft^
by this method, the proportions of fpheric and
cylindrip fuperficies to one another, by know-
ing the folid content; and on the qther hand,
the folid content, by kpowipg the fuperficies,
with admirable facility, and mpft full fatisfa-
|tion in thofe things which are rigidly ga«
ther'd by pure Geometry*

2. Let vi fuppofe a feries of .quantities to pro*
teed from o (inclufive) in a duplicate Arithmetic
t\ognJJiQnt that is, p, 1, 4, 9, 16 > &c. thefquares

t *o6 J

of numbers in afmple Arithmetic progpejflon, o, !,
2, 5, 4, &c. And the triple of this J cries will
always exceed the greateft term multiplied by the
number of terms ; but the number of terms 771-
creafing the proportion, continually approximates
tii at laft it comes to an equality when the num-
ber of terms is increased in infinitum.

5x0-4-1=3. j

2X1 S5E 2. z

5x0-4-1-4-4^=1 J. 15 I
1 5 x 4 55 12. 12 4*

3x 0-4-1 -4-4-4-^=42. 42 7^

4 x 9 = 56. 56 6
3x0*4-1-4-4-4-9-+ 16 ^90. 9© _9

S x 16 35 80. 8o — |"

5x0-4-1 -4. 4-4-9-4- ii-»-^j=ri6j.i6j ijt

6x 2$ = 125.11$ 10

As for example, if the terms be two, the
triple of the terms will be to tlie greateft
term drawn into the number of terms as 5 to
2 ; if there be three terms as $ to 4 • if tour,
as 7 to 6 ; if five, as 9 to 8, and fo conti-
nually : So that the antecedents of thefe pro-
portions always mutually exceed one another
by the number 2 $ and fo every antecedent its
consequent by 1. Whence it is evident that
by how much the greater the number of terras
is, by fo much the more the proportion tends
to equality. So 100 to 99 is lefs diftant from
the proportion of equality than tm to 9.
From hence, fuppofing the number of terms
Infinite (or infinitely great,) the triple of quan-
tities proceeding, thus in a duplicate propor-
tion (or as the Iquares of the numbers, o, r ,
*9 ?» 4t &V. will be equal to as many quan-
tities equal to the greateft term*

The

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C r°7 3

The fame, as to the fubftance of it, is laid
down by Archimedes in his Book of Spirals, as
the Foundation of many Argumentations, in
that, and other Books, and is weil demonftra-
ted by our Learned Country-man Dr. IPattu :
However, I thought fit to illuftrate the irui-
ter by this method, as being not unworthy
our Confideration, and very perfpicuous and
intelligible in this, that 'tis free from Fra-
ctions. And by the way 'tis obferv'd, that
from hence we may eafily find the proportion
of a feries triple to as many terms equal to
the greateft, viz. as twice the number of terms
lefs one, to twice the number of terms lels
two. So that if the number of terms be 6%
the proportion of a feries triple to as many
terms equal to the greateft will be as it.

It will be a very eafy and apt Illuftration
of this Rule, if we infer hence, That a Cone is
fubtriple of a Cylinder, having an equal bafe
and altitude. For let us fuppofe the altitude
AE of the cone ZY to be divided into equal
and indefinitely many parts, by as many pa-
rallel right lines ZY, and the lines ZY will
be as the numbers i, 2, 5, 4, gfo and the
fquares or circles conftituted upon the diame-
ters ZY, as r, 4, 9, *<5> whence all thofe
circles, or the whole cone AZY (made up

of the fame) will be fub-
triple of as many circles e-
quai to the greateft, conftitu-
ted on the greateft diameter
ZEY, that is, fubtriple of a
cylinder whofe bafe is AEY,
and altitude AE.
There occur two other irioft apt examples
of this Rule, viz. by inferring, That the compler
vient of a Semiparabola is fubtriple of a paraUe-
logram having the fame bafe and hcighth ; as

alfo?

1

Google

C SoS ]

alfo, That the Jpacc comprehended by the Spiral
and Radius it fubtriple of the circle in which
the 'Spiral is generated ; But of thefe in ano-
ther place. Wherefore to go on with what
we began, thefe two Rules being fuppofed ;
let us conceive ZAY to be a fegment of a

fphere, X its center,
AT its diameter, and
ZAYT a great circle
palling thro' the ver-
tex, and the part AE
of the Axe to be di-
vided into an indefi-
nitely many equal parts 5
and let us imagine pa-
rallel lines to be drawn
thro' the points of Di»
vifion, generating cir-
cles in the fphere,
whofe Radii let be BZ, CZ, DZ, and diame-
ters ZY. I fuppofe the fegment of a fphere
to confift of all thefe parallel circles, whole
number is as great as that of the points, or
equal indefinitely many fmall parts in the Axe
AE, aecording to the known Method of In*
divijibles.

But now for brevity's fake, let the dia-
meter AT be called d, and the radius of the
fphere r (if need be,) and the Axe AE, by
which the number of terms is exprefs'd, call
*f, and one of the equal parts a ; which be-
ing fettled, 'tis evident, (by the Elements) that

SZ1 = ABxBT=ax"<i ad — a\ and

in like manner CZ* = AC x CT = z a x

d — z a — z ad — 4 a1, and by the fame rea-

foningDZ* =rADxDT= 3 ad — gax9 andEZ*
=AE x ET= 4 tut — 16 a*9 Sec. that is, that
the tquares of the radii of the circles ZY are

to

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I f<>9 D

to one another as the re&angles ad, zadt ^ad,
Ajd9 Sec. (which proceed in an Arithmetical
Progrefliort from o) lefs by the fquare a1, 4a1,
pj% i6a%, Sec. which go on as the fquares
of the numbers, 1, 2, j, 4, &cJ) But by our
firft Rule, all the Reftangles o, ad. zad, $ai9
qad, Sec. are equal to half as many terms
equal to the greateft AE x AT or ndf that is,
nd x n

Moreover, by our fecond Rule* all the
fquares o, a*9 401, 94s, i6a*9 fijV. taken to-
gether, are equal to a third part of as many

terms equal to the greateft AE1 or »Y, that

n* x n

. Wherefore all the fquares defcribed upon
the radii BZ, CZ, DZ, EZ, conjunctly, are

equal to the difference , (or the

terms being redue'd to the fame denomination,).

3 72 g-x nnn ^ ^4^^ that ^ ^

the fquares defcribed upon the diameter ZY,

. 12 ndn~ 8»* 6 win — 4 nl
are equal to g or — ^— — V

Whence a fegment of a fphere is equal to a

S Under, the diameter of whofe bafe is the
5 of a fquare equal to 6 -nd — 4 n1, add alti-
tude is f ?! ; or to a cone having the fame bafe,
but the altitude 71, or which is all one, having

3 bafe whofe radius is ^/ ~- ^— • or *J

\ *nd — n1, and altitude n as before. Which
Cone we may change into a Cone upon the
fame bafe ZY with the fegment ZAY, by

faying, as ZE* (?, e. dn - %*) to | »i - ** or

(both

froth terms being divided by n) is d — nth
X d — n, fo reciprocally n to the altitude of
the Cone fought: Or in the figure by ma-

For ES mil he the altitude of the Cone ZSY e-
qual to the fegnient of the fplme ZAT* Which
is a noted Theorem of Archimedes demon-
ftrated by him with fo much labour and

prolixity. .

Hence, if the given legment be a Hem-
fphere^ and io n — ~ dor r9 then d or z r will
be the altitude of a Con*, tfrhich having a
bafe equal to the bafe of the Hemifpbere% (or
to the greateft circle in the fphere,) will be
equal to the Hemifphere* And a Cone whofe
bafe is double of the greateft circle, and the
altitude * r, or the Cylinder whofe bafe is V
of the greateft circle* and altitude i r, will
be equal to the whole Sphere* Whence the
whole Sphere is f Of a Cylinder the diameter
of whole bafe is z r, and the altitude alfo z U
And this is the chief Theorem of Archime-
des % viz. That & fphere it fubfefquialter or -f-
of that Cylinder , wnoje Altitude and Diame-
ttr of the bafe is ejual to the Diaincttr of the

Sphere. ,

Furthermore, not to pafs over any thing
in our Author which feems to be to our pur-
pofe :

If to the furri firft found, representing a feg~
. 6ndn— Annn, weaddx^n— 6dn%-+An*

mnt&z.—^ — — ?

(^dn- 4* x i-±m ^ f m x XE) ^

Tenting the Cone ZXY* the aggregate f ddn
will reprefent the Seftor of the Sphere ZX-
YA, which for that reafon will be equal to
a Cylinder, the diameter of whofe bafe *J dn%

ana

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ahd the altitude | d, or to a Cone, the dia-
meter of whofe bafe is j and the altitude
zd, or alfo in a Cone, the Radius of whofe

bafe is V in* and the al^itudc t ^ = r (it be-
ing reciprocally as tfn : dn :: zd: ± df) that is,
to a Cone, the Radius of vfhofe bafe is the
Line AZ, drawn from the vertex to the
circumference of the bafe of the fegment, (for

AZT = TA x AE = dny) and the altitude r.
And this is the next famous Theorem of Arch-
viedesy concerning the folidity of the feftor
of the Sphere, viz. That the feBor of a fpben
is equal to a Cone, whofe bafe is a circle ie-
fcribed by a Radius equah to a line drawn from,
tke vertex to the circumference of the bafe of the
fegment, and whofe altitude is equal to the Ra-
dius of the fphere.

And thus I think I have completed that
which belongs to the folidity of a fphere, and
its parts with fufficient brevity and perfpi*
cuity. From hence we fhall deduce the Re-
folution of the other Problem, which I pro-
pofed concerning the Surface of the fegment
of a fphere ; and then of the whole fphere*
To obtain this, as we fuppofedj before a Cir-
cle to confift of concentric Peripheries, and
the Se8or of a Circle, of concentric Arcs, (in
the number of which, the greateft, and the
leaft or a point is reckon'd : So now we
fuppofe fpheres to confift of concentric fphe-
ricai fuperficies, and the SeSors of Spheres, of

like concentric fuperfi-
cies ; as for example, the,
feftor of the fphere ZAE,
of the fupeificies BZ, CZf
DZ, EZ, &c.} which
fuppofition indeed feems
% fo eafy and natural, that
in my judgment 'tis fuffi-

C 3

cient only to propofe it 5 neither Is a furi
tfcer explication wanting to gain an affent
to it

2. We fuppofe thefe fpherical fupetficies to
be in a duplicate Ratio of the Radius of ~
the fpheres : This is the common affeftion
of all like fupetficies, and it feems to agree
very well with the fupetficies of fpheres, De-
caufe they appear to be moft uniform and fi-
milar. But this Suppofition might eafily be
evinc'd and eftablifh'd by the fame loft of
arguing, as fpheres are proved to be in tri*

Slicate proportion to their Diameters or Ra-
ii ; Or might have been join'd as a Corollay
to Prop. 17. and 18. Elm. n. where the fu-
perficies of like Polygones are fupposM to be
infcribed in fpheres, having as well the fuper-
ficies in a duplicate, as the folidity in a tri*
plicate Ratio of the Diameters of the Spheres*
Thefe things being premis'd, let us fuppofe
AE a Radius or the fide of the Seftor of a
Sphere EAZ,' to be divided into equal and
indefinitely many fmall parts, and the feftpr
AEZ to confift of thele fpherical fuperficies
BZ, CZ, DZ, EZ, it will be evident that all
thofe fuperficies in the Progreffion are as the
fquaresof the Radii, that is, as AB1, AC*,
AD% AE% &c. or as the fquares of the num-
bers 1, 2, 3, 4, &c. whence by out fecond Rule,
the fum of all thefe fuperficies, that is, the
feftot AEZ, will be \ of as many fuperficies
equal to the greateft FZ, that is, { of the
greateft EZ, drawn into r the number of terms.
Whence a feftor is equal to a Cylinder whole
bafe is | of the greateft or extreme fuperficies
of the feftor, and whofe altitude is r : Or to
a Cone whofe bafe is equal to the fuperficies
of the feftor, and its altitude r, which is the
laft oi Lib* 1.) but we juft now prov'd that a

feftor

1

1

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C j

fcftor is equal to a Cone whofe altitude is rf
and bafe a circle defcrib'd by the Radius YE,
drawn from the vertex of the fegment EYZ to
the circumference of the bafe. Wherefore, a
Cone whofe altitude is, and bafe equal to the
fuperficies of the feftor, is equal to a Cone of
the fame altitude, whole bafe is a circle de-
fcrib'd by the Radius YE.

And fo the fuperficies of the feftor EYZ is
equal to a circle defcrib'd by the Radius YE*
Which, certainly is the principal Theorem of
all thofe that occur in the Books of Avchi-
medes, nor is there found a more excellent
pne in all Gteoijietry ; viz. That the fupetficies
of any fegment of a fphere is equal to a emit
whofe Radius is a right line drawn horn the ver?
tex of the fegment to the circumference of the
iafes : And hence, That the fuperficies of an He*
mifphere is double of the bafe9 or equal to twa
great circles of the fphere.

For in this Cafe YE2 = A^1 AY1 s z

AE1 and confequently a circle defcribed by

the Radius YE is equal to
two circles defcrib'd by the
Radius AE. Whence alfo,
the fuperficies of the whole
fphere is quadruple a circle
laving tl*e fame Radius with
the fpherey that is, quadruple
the greatefi circle in the fphere $ and equal to a
circle wJjofe Radius is the diameter of the fphere*
From hence it follows, That the fuperficies of a
fphere is equal to the fuperficies of a Cylinder of
the fame heighth and breadth ; for the fuperfi-
cies of that Cylittder is quadruple to the bafe*
as we ihall fhew hereafter. And thefe are the
inoft noted Theorems of Archimedes. Nay,
from hepce all thofe things follow, which he

K k hat

c rii 3

has written concerning the fuperficies of
fpheres, and their fegments. So that from
thefe few and eafy Suppofitiotis, I have demon-
itrated whatever feem to be of any Note in
the Books Of the Sphere and Cylinder.

I will only add, that after by the method
of Jrchimedesy (for I think fcarce any other
can be invented, befides ours , for finding
the folidity) the fuperficies of fegments are
found equal to the circle defcribed by the
Radii YE ; hence it will plainly follow, that '
the fuperficies of fpheres, and thence of like
feftors are in a duplicate ratio of the Radii
of the fpheres 5 and consequently from the
fuperficies thus found, the contents of feg-
ments, and of whole Tpheres may be mutually
deduced, and that very qlearly and expedi-
tioully after this manner. Becaule in the le-
ftoi EAZ (fig. Pag. 558.) the fuperficies BZ,
CZ, DZ, EZ, proceed as the fquares defcrib'd
upon AB, AC, AD, AE, that is, as t, 4, *o,i6,
&c. the whole fecior will be equal to J of as
many fuperficies equal to the greateft EZ, or
EZ x r, that is, to a Cylinder whofe*/bafe is
I EZ, and altitude t\ or to a Cofte whofe
pafe is t Z, and altitude & But £Z is fup^o-
ied equal to a circle whofe Radius' is YE,
wherefore the feftor EAZ is equal to a Cone
whole bale is a circle dsfcribed by the Ra-
dius YZ and altitude r : Which is Apchime-
tlcs^s univerfal Theorem for the contents of
Circles. Whence if from this the Cone ZAE
Handing on the bafe of the fegment EYZ, .
and having the vertex at the center of tJie
fphere A, be lubdu&ed, (yo^Tll have that feg-
ment EYZ.) But when the ietfor EYZ is a
.Hemifphere, there will be no fuch Cone to
be lubduded ^ and for that reafort a Cylin-
der whole bafe is | EZ, and altitude r, or

the

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[ pi ]

the Cone whofe bafe is *EZ, and altitude
likewife r will be equal to the whole fphere.
But the fuperfieies of the Hemisphere EZ, is
proved to be equal to two of the greateft
circles in the fphere, whence the whole fphere
is given. This is Jrcbimedes's firft and prin-
cipal Theorem, for the content of a fphere ;
whence 'tis eafily deduced, that a fphere is £
of a circumfcrib'd Cylinder, that is, of a Cy-
linder whofe altitude and diameter of its bafc
is equal to the diameter of the fphere.

The Do&rine of our other [Archimedes]
feems to make agaiuft, and fubvert the new
and celebrated Method of Indivifibies, and is
prefs'd to that end by Tacquet^ for inftance,
(Prop. z. lib. 2. Cylindr.) For the ufual procefs
of that method feems to exhibit the diroefl-
fion of the fuperfieies of a Cone, (as aJfo of
a fphere, and of other Curves) different enough
from what pur Author and others have de-

monftrated : As for exam-
ple, let us fuppofe ABCD
a right Cone, whofe Axe
is AX, and bafe BCD, and
plane . 0 x ^ drawn, at
pleajure, parallel to the
bale BCD. And fince, as
Diam. BD : Peripb. BCD ::
Diam. (if : Peripb.
and fo every where it will
be (according to the Method
of Indrjifiblesy and by 12. 5.) as Diam. BD,
to fieiipl. BCD,, fo istJie triangle ABD, con-
fitting of thofe ' parallel Diameters, to the
Conic Superficies ABCD* coniifting of thofc
Peripheries, i.e. Diam* BD : Peripb. BCD

AX x BD : AX x Penph. BCD ... .

— ■■■ A 1 Whence

2 z •

X k z AX

• — will be equal to the fuper-

ficies of the Cone ; fchich is falfe and contra-
ry to -what was defnonft rated juft now. For
we demonfttated that the JTuperficies " "

Cone was

V 1 — '

2

In anfwering this Ob'ieftion, we fay, that
the Method of Indivijibks, in the f peculation of
Perimeters, and of Curve Surfaces, proceeds
other wife than in the fpeculation of plane Sur-
faces and folid Contents. It does indeed fup-
pofe that the Area of plane Figures confiit,
as it were of parallel right lines, and the con-
tents of fblids of parallel Planes, and that
their number may be exprefs'd by the altitude
of the Figures : But it by no means fuppofes,
that the Perimeters of plane figures connft of
points, or the fuperficies of fol ids, of lines, the
number of whicn may be exprefs'd by the al-
titude of the figure. As for example, akho'
the triangle ABD (in the laft figure) confifts
of lines parallel to BD, the number of w liic Ii
is expreiled by the number of points in the
perpendicular AX, that is, by the length of
the perpendicular : Yet it would be ablurd to
fuppofe that the line AB confifts ef pointy
whole number may be eiprefs'd by ,the uum- 4
ber of points in a lefs line AX. Fowaltho*
the right line /32 dra\im thro* each infinitely
fmall part of AX, divide AB into, as many
infinitely fmall Darts, yet thofe, parts are not
of the fame Denojnination or Quality with
the parts of AX, but fomewhat greater than
them y fo that if the pahs of AX bi look'd
upon as points, the .parts of AB are irb't fto be
called points, but greater thafr* points.* and
on the contrary, if the parts of AB be called
- ** ♦ «f poinrst

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w m

C ]

points, the parts of AX are to be look'd upon
as Ms than points, if it be lawful to fpeak
fo. For the points which are treated of in
the Method of Indivijibles are not absolutely

f>oints% but indefinitely fmall parts, which u-
urp the names of points, becaufeof the Affini-
ty. Since therefore points don't admit of great-
er and lefs, the name of points is not at the
fame time to be attributed to the parts of diffe-
rent magnitudes ^ confequently tho* the number
of the greater parts of AB maybe exprefs'd by
the number of the leflcr parts of Aa, yet the
number of points in ABcan no ways oe ex-
pfeiTed by the number of points in AX, (that is,
t>y the number of parts in AX, equal to the
number of parts in AB, which are called points.)
The line AB Iras as many points as there are in
it felf alone, or another line equal to it felf,noc-
can it be determin'd by any other meafure. Af-
ter the fame manner, this method don't fuppofe
the conic Surface ABCD to confift of as many
parallel circumferences perpetually ijicreafing
from the vertex A, or decreafing from the bafc
BD, as there are points in the Axe AX ; but ra-
ther of as many thus increafing or decreafing as
there are points in the fide AB. For in the
Revolution of the line AB about the Axis AX,
(whereby the fuperficiesof the Cone is generated)
every point in the line AB produces a circum-
ference, and confequenrly more circumferences
are produced than the points contained in the
Axis AX. Therefore if you would extend the
Method of bidivifilrles to the fupcrficies of folid.%
and fuppofe thofe fuperficies to confift of paral-
lel lines, ycu ought not to compute this by the
parallel Areas conftituting the iolid, that is, not
to number thofe Areas by the altitude of the lo-
lid, but by other lines agreeable to the condition
of each figure. Which lines, in figures that are
net irregular, may eafily be dettnnin'd : For in-

ftance,

C f 18 ]

ftance, in the equilateral Pyra-
mid ABCD, whofe Axe is AX,
fuppofing that the lateral furface
of the Pyramid confifts of Peri-
meters of triangles, parallel to
the bafeBCD, thefe can neither
be computed by the altitude
AX, nor by the iide AB, (for by the former,
the thing requir'd, would be wanting of the true
Dimenfion, and by the latter 'twould exceed it,)
but by the line AE draw n from the vertex A per-
pendicular to the fide EC of the bafe ; thereafon
of which is, that every plane fide of a Pyramid
as ABC, confiih of parallel right lines computed
by the altitude AE. After the fame manner,
fuppofingthat the fuperficies of the Hemifphere

BAD, confift;of peripheries
of circles parallel to the
0 bafe BCD, the number of
them is not to be compu-
ted by the Axis AX, but
by the Quadrantal Arc AB,
C becaufe that every point of

the Arc AD in revolvjug produces a circumfe-
rence. And fo any fuperficies, whether plane or
curv'd, which is conceived to confift of equidi-
ftant right or curv'd lines, is to be computed
by a line cutting thofe equidiftant lines perpen-
dicularly. For hnce thofe equidiftant lines, in
this Met bod of Indivifibles, are notconfider'd ab-
solutely as lines having an infinitely fmall breadth,
which is the fame with the breadth or thicknefs
of the point defcribing thofe equidiftant lines in
their Circumvolution, andfince the farpe equi-
diftant lines divide the line cutting them per-
pendicularly into parts meafuring its, breadth,
thofe parts are to be look'd upon as luch fort of
points, and confequently the number of equidi*
ftant lines, or the fum of thofe breadths is to be
computed by the number of points in the line..

cutting

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C y*9 3

cutting them perpendicularly, that is, by tht
length of that line, and not by a line of any
other length, for that will coiifcft of more Of
lefs points, "fj*

Hence therefore in the (peculation of the fu-
petficiesof folids, the 'Method qf Indivtptts is
not unufeful, but rather, very commodious/ pro-
vided it be rightly underftood, and applied a^
cording to the Rule prefcribM. 4For ft the
helpofit eyfeo di<db^.^iesavia7te'%«tf^
if lb be* weV have fome convenient Data pre-
fuppos'd, on whidh the reafonlng may be found-
ed. Forinftarice, we might by the help of It,
hiveftigate the fuperflcies of a Cone, by r£a-
oning after this manner. „

If the fiiperficies of the cohb AB(j (fig.pag. 5 6*0
be divided into innumerable Peripheries of cir-
cles 0xfi parallel to the fraife BCD, the breadth
of thole Peripheries taken toother, make up the
fide AB cutting them perpendicularly, and con-
fequently there will be as many Peripheries as
there are poiats in the line AB, that is, their
number may be exprefs'd by the number of
points in AB, or by its length. 'Wherefore, if
you draw perpendiculars equal to the Peripheries
to every point of AB, a,fuperficies will be fhade
out of tnofi perpendiculars equal to thefupef-
ficies of the Gone. B\it< that: fuperficies will be
a triangle wlfofe heighth is AB, and bafe^qual
to the greateft Periphery BDC, and fo the fu-
perficies of -thcrCone wftl be ^ 4- AB x Perif%.
BDC, which conclufioa agrees with the things
laid down and demonftlated by Archimedes.^

After the lame manner, if you take any right

• - *>8 equal to

the quadrant al A rc
AB of the Hemi-
fphere(i8/W£.$6<«.)
and to each of its
points p let the

pei««

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[ ]

perpendiculars ^ J>e erpftcd equal to the Radii
JIN of parallel circles MOM palling thro* the
correfponding points M of that quadrantal Arc,
the greateft of which let be equal to the Ra-
dius BX of the bafe ot the Hemifphere : The
figure ct££ will contain the Radii of all the cir-
cles of whofe Peripheries the fuperficies of the
ifphere confifts. And if the perpendicular

ft, 0f be ere£le& eoual to the Peripheries
DM, BDB, there will be made the figure ctfif
equal to the fuperficies of the Ilemifphere. The 0
dimenfion of which figure if you can by any
means find (as in this cafe you are to find the
Area of the figure «jgg) thence you will eafily
deduce the content of the fegment of the fphere,
agreeing to what you may gather by any law-
ful reasoning. Which Obfervation, I think
will not be unufeful in Geometry.

FINIS,

f.

AT the Hand and Pen in Bar-
Mean are Taught, ©aritfaff,

Common and sajBtc&antjs accounts,

after that well approved Method of
Mr. KormaAfcl, aiffCUja, Geometry,

$&eaftttm& Surveying,Gauging,Ba-
tfgatfon and Dialing, with other
parts of the ^atfcemattcltf, alfo thp
Ufe of the$IobC& and other

Inftruments, by me

Robert Arnold,

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1 NUN

CIRCULAl

A 543960

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